Reconstruction of Modular Data from ${\text {SL}}_2({\mathbb {Z}})$ Representations

Abstract

Modular data is a significant invariant of a modular tensor category. We pursue an approach to the classification of modular data of modular tensor categories by building the modular S and T matrices directly from irreducible representations of ${{\text {SL}}_2({\mathbb {Z}}/ n {\mathbb {Z}})}$. We discover and collect many conditions on the ${{\text {SL}}_2({\mathbb {Z}}/ n {\mathbb {Z}})}$ representations to identify those that correspond to some modular data. To arrive at concrete matrices from representations, we also develop methods that allow us to select the proper basis of the ${{\text {SL}}_2({\mathbb {Z}}/ n {\mathbb {Z}})}$ representations so that they have the form of modular data. We apply this technique to the classification of rank-6 modular tensor categories, obtaining a classification of modular data, up to Galois conjugation and changing spherical structure. Most of the calculations can be automated using a computer algebraic system, which can be employed to classify modular data of higher rank modular tensor categories. Our classification employs a hybrid of automated computational methods and by-hand calculations.

Appendices

List of ${\text {SL}}_2({\mathbb {Z}})$ Irreducible Representations of Prime-Power Levels

In this section, we list all the ${\text {SL}}_2({\mathbb {Z}})$ symmetric irreducible representations of dimension 1–6, whose level ($l={\text {ord}}(\rho (\mathfrak {t}))$) is a power of single prime number, which are generated by the GAP program [27]. In the list, $\rho (\mathfrak {t})$ is presented in term of topological spins $({\tilde{s}}_{1},{\tilde{s}}_{2},\cdots )$ (${\tilde{s}}_{i} = \arg (\rho _a(\mathfrak {t})_{ii})$).

Note that $\rho (\mathfrak {s})$ is symmetric and $\rho (\mathfrak {s})_{ij}$’s are either all real or all imaginary. When $\rho (\mathfrak {s})_{ij}$’s are all real, $\rho (\mathfrak {s})$ is presented as $(\rho _{11}, \rho _{12}, \rho _{13}, \rho _{14}, \cdots ;\ \ \rho _{22}, \rho _{23}, \rho _{24}, \cdots )$. In this case, $\rho (\mathfrak {s})^2={\text {id}}$ and the representation $\rho$ is said to be even. When $\rho (\mathfrak {s})_{ij}$’s are all imaginary, $\rho (\mathfrak {s})$ is presented as $\textrm{i}(-\textrm{i}\rho _{11}, -\textrm{i}\rho _{12}, -\textrm{i}\rho _{13}, -\textrm{i}\rho _{14}, \cdots ;$ $-\textrm{i}\rho _{22}, -\textrm{i}\rho _{23}, -\textrm{i}\rho _{24}, \cdots )$, or as $(s_n^m)^{-1}(s_n^m \rho _{11}, s_n^m \rho _{12}, s_n^m \rho _{13}, s_n^m \rho _{14},$ $\cdots ;$ $s_n^m \rho _{22}, s_n^m \rho _{23}, s_n^m \rho _{24}, \cdots )$, where $s_n^m:= \zeta _n^m-\zeta _n^{-m}$. In this case, $\rho (\mathfrak {s})^2=-{\text {id}}$ and the representation $\rho$ is said to be odd. In any case, the numbers inside the bracket $(\cdots )$ are all real. We can tell a representation to be even or odd by the absence or the presence of $\textrm{i}$ or $(s_n^m)^{-1}$in front of the bracket $(\cdots )$.

We note that two symmetric representations are equivalent up to a permutation of indices, and a conjugation of signed diagonal matrix. To choose the ordering in indices, we introduce arrays $O_i =[\text {DenominatorOf}({\tilde{s}}_{i}),{\tilde{s}}_{i}, \rho _{ii}]$. The order of two arrays is determined by first comparing the lengths of the two arrays. If the lengths are equal, we then compare the first elements of the two arrays. If the first elements are equal, we then compare the second elements of the two arrays, etc. To compare two cyclotomic numbers, here we used the ordering of cyclotomic numbers provided by GAP computer algebraic system. We order the indices to make $O_1\leqslant O_2 \leqslant O_3 \cdots$. The conjugation of signed diagonal matrix is chosen to make $-\rho (\mathfrak {s})_{1j} < \rho (\mathfrak {s})_{1j}$ for $j=2,3,\ldots$. If $\rho (\mathfrak {s})_{1j}=0$, we will try to make $-\rho (\mathfrak {s})_{2j} < \rho (\mathfrak {s})_{2j}$, etc.

All the prime-power-level irreducible representations are labeled by index $d_{l,k}^{a,m}$, where d is the dimension and l is the level of the representation. The irreducible representations of a given d, l can be grouped into several orbits, generated by Galois conjugations and tensoring of 1-dimensional representations that do not change the level l: the k in $d_{l,k}^{a,m}$ labels those different orbits. If there is only 1 orbit for a given d, l, the index k will be dropped.

The irreducible representation labeled by $d_{l,k}^{a,m}$ is generated from the irreducible representation labeled by $d_{l,k}^{1,0}$ via the following Galois conjugations and tensoring of 1-dimensional representations

$$\begin{aligned} \rho _{d_{l,k}^{a,m}}(\mathfrak {t})&= \sigma _a\big (\rho _{d_{l,k}^{1,0}}(\mathfrak {t}) \big )\textrm{e}^{2\pi \textrm{i}\frac{m}{12}} \nonumber \\ \rho _{d_{l,k}^{a,m}}(\mathfrak {s})&= \sigma _a\big (\rho _{d_{l,k}^{1,0}}(\mathfrak {s}) \big )\textrm{e}^{-2\pi \textrm{i}\frac{m}{4}} \end{aligned}$$

(A.1)

where the Galois conjugation $\sigma _a$ is in ${\text{ Gal }}({\mathbb {Q}}_n)$ with n be the least common multiple of ${\text {ord}}(\rho _{d_{l,k}^{1,0}}(\mathfrak {t}))$ and the conductor of $\rho _{d_{l,k}^{1,0}}(\mathfrak {s})$. The Galois conjugation $\sigma _a$ is labeled by an integer a, which is given by

$$\begin{aligned} \sigma _a\big (\textrm{e}^{2\pi \textrm{i}/n} \big ) = \textrm{e}^{2\pi \textrm{i}a/n} . \end{aligned}$$

(A.2)

Also $m \in {\mathbb {Z}}_{12}$ is such that ${\text {ord}}(\rho _{d_{l,k}^{1,0}}(\mathfrak {t})\textrm{e}^{2\pi \textrm{i}\frac{m}{12}})= {\text {ord}}(\rho _{d_{l,k}^{1,0}}(\mathfrak {t}))$. Due to this condition, when l is not divisible by 2 and 3, m can only be 0. In this case, we will drop m. Here we choose $d_{l,k}^{1,0}$ to be the representation in the orbit with minimal $[{\tilde{s}}_1, \tilde{s}_2,\cdots ]$.

The numbers of distinct irreducible representations with prime-power level (PPL) in each dimension are given by

In the above we also list the numbers of distinct irreducible representations, which are tensor products of the irreducible representations with prime-power levels.

In the following tables, we list all irreducible representations with prime-power levels for rank 2, 3, 4, 5. For rank 6, to save space, we only list all irreducible representations with prime-power levels that have a form $\rho _{d_{l,k}^{1,0}}$. Other irreducible representations, with prime-power levels and the same dimension, can be obtained from those listed ones via Galois conjugations and tensoring 1-dimensional representations. In the Supplementary Material section of the arXiv version of the article we list all distinct irreducible representations of prime-power levels. In the tables $c_n^m:= \zeta _n^m + \zeta _n^{-m}$ and $s_n^m:= \zeta _n^m - \zeta _n^{-m}$.

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$1_{1}^{1}$	1	(0) ,   (1)
$1_{2}^{1,0}$	2	$( \frac{1}{2} )$,   ($-1$)
$1_{3}^{1,0}$	3	$( \frac{1}{3} )$,   (1)
$1_{3}^{1,4}$	4	$( \frac{2}{3} )$,   (1)
$1_{4}^{1,0}$	5	$( \frac{1}{4} )$,   $\textrm{i}$(1)
$1_{4}^{1,6}$	6	$( \frac{3}{4} )$,   $\textrm{i}$($-1$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$2_{2}^{1,0}$	1	$( 0, \frac{1}{2} )$,   ($-\frac{1}{2}$,$-\frac{\sqrt{3}}{2}$; $\frac{1}{2}$)
$2_{3}^{1,0}$	2	$( 0, \frac{1}{3} )$,  $(s_{3}^{1})^{-1}$(1, $-\sqrt{2}$; $-1$)
$2_{3}^{1,8}$	3	$( 0, \frac{2}{3} )$,  $(s_{3}^{1})^{-1}$($-1$, $-\sqrt{2}$; 1)
$2_{3}^{1,4}$	4	$( \frac{1}{3}, \frac{2}{3} )$,  $(s_{3}^{1})^{-1}$(1, $-\sqrt{2}$; $-1$)
$2_{4}^{1,0}$	5	$( \frac{1}{4}, \frac{3}{4} )$,  $\textrm{i}$($-\frac{1}{2}$, $\frac{\sqrt{3}}{2}$; $\frac{1}{2}$)
$2_{5}^{1}$	6	$( \frac{1}{5}, \frac{4}{5} )$,  $(s_{5}^{1})^{-1}$(1, $-\frac{1+\sqrt{5}}{2}$; $-1$)
$2_{5}^{2}$	7	$( \frac{2}{5}, \frac{3}{5} )$,  $(s_{5}^{2})^{-1}$(1, $\frac{1-\sqrt{5}}{2}$; $-1$)
$2_{8}^{1,0}$	8	$( \frac{1}{8}, \frac{3}{8} )$,  ($-\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{\sqrt{2}}{2}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$2_{8}^{1,9}$	9	$( \frac{1}{8}, \frac{7}{8} )$,  $\textrm{i}$($-\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{\sqrt{2}}{2}$)
$2_{8}^{1,3}$	10	$( \frac{3}{8}, \frac{5}{8} )$,  $\textrm{i}$($-\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{\sqrt{2}}{2}$)
$2_{8}^{1,6}$	11	$( \frac{5}{8}, \frac{7}{8} )$,  ($\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{\sqrt{2}}{2}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$3_{3}^{1,0}$	1	$( 0, \frac{1}{3}, \frac{2}{3} )$,  ($-\frac{1}{3}$, $\frac{2}{3}$, $\frac{2}{3}$; $-\frac{1}{3}$,$\frac{2}{3}$; $-\frac{1}{3}$)
$3_{4}^{1,0}$	2	$( 0, \frac{1}{4}, \frac{3}{4} )$,  (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$,$\frac{1}{2}$; $-\frac{1}{2}$)
$3_{4}^{1,3}$	3	$( 0, \frac{1}{2}, \frac{1}{4} )$,  $\textrm{i}$($-\frac{1}{2}$, $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{\sqrt{2}}{2}$; 0)
$3_{4}^{1,9}$	4	$( 0, \frac{1}{2}, \frac{3}{4} )$,  $\textrm{i}$($\frac{1}{2}$, $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{\sqrt{2}}{2}$; 0)
$3_{4}^{1,6}$	5	$( \frac{1}{2}, \frac{1}{4}, \frac{3}{4})$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$;$\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{5}^{1}$	6	$( 0, \frac{1}{5}, \frac{4}{5} )$,   ($\frac{\sqrt{5}}{5}$, $-\frac{\sqrt{10}}{5}$, $-\frac{\sqrt{10}}{5}$; $-\frac{5+\sqrt{5}}{10}$, $\frac{5-\sqrt{5}}{10}$; $-\frac{5+\sqrt{5}}{10}$)
$3_{5}^{3}$	7	$( 0, \frac{2}{5}, \frac{3}{5} )$,   ($-\frac{\sqrt{5}}{5}$, $-\frac{\sqrt{10}}{5}$, $-\frac{\sqrt{10}}{5}$; $-\frac{5-\sqrt{5}}{10}$, $\frac{5+\sqrt{5}}{10}$; $-\frac{5-\sqrt{5}}{10}$)
$3_{7}^{1}$	8	$( \frac{1}{7}, \frac{2}{7}, \frac{4}{7} )$,   ($-\frac{c_{28}^{1}}{\sqrt{7}}$, $-\frac{c_{28}^{3}}{\sqrt{7}}$, $\frac{c_{28}^{5}}{\sqrt{7}}$; $\frac{c_{28}^{5}}{\sqrt{7}}$, $-\frac{c_{28}^{1}}{\sqrt{7}}$; $-\frac{c_{28}^{3}}{\sqrt{7}}$)
$3_{7}^{3}$	9	$( \frac{3}{7}, \frac{5}{7}, \frac{6}{7} )$,   ($-\frac{c_{28}^{3}}{\sqrt{7}}$, $-\frac{c_{28}^{1}}{\sqrt{7}}$, $\frac{c_{28}^{5}}{\sqrt{7}}$; $\frac{c_{28}^{5}}{\sqrt{7}}$, $-\frac{c_{28}^{3}}{\sqrt{7}}$; $-\frac{c_{28}^{1}}{\sqrt{7}}$)
$3_{8}^{1,0}$	10	$( 0, \frac{1}{8}, \frac{5}{8} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{8}^{3,0}$	11	$( 0, \frac{3}{8}, \frac{7}{8} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{8}^{3,3}$	12	$( \frac{1}{4}, \frac{1}{8}, \frac{5}{8} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{8}^{1,3}$	13	$( \frac{1}{4}, \frac{3}{8}, \frac{7}{8} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{8}^{1,6}$	14	$( \frac{1}{2}, \frac{1}{8}, \frac{5}{8} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{8}^{3,6}$	15	$( \frac{1}{2}, \frac{3}{8}, \frac{7}{8} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{8}^{3,9}$	16	$( \frac{3}{4}, \frac{1}{8}, \frac{5}{8} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{8}^{1,9}$	17	$( \frac{3}{4}, \frac{3}{8}, \frac{7}{8} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{16}^{1,0}$	18	$( \frac{1}{8}, \frac{1}{16}, \frac{9}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{16}^{7,3}$	19	$( \frac{1}{8}, \frac{3}{16}, \frac{11}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{16}^{5,6}$	20	$( \frac{1}{8}, \frac{5}{16}, \frac{13}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{16}^{3,9}$	21	$( \frac{1}{8}, \frac{7}{16}, \frac{15}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{16}^{5,9}$	22	$( \frac{3}{8}, \frac{1}{16}, \frac{9}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{16}^{3,0}$	23	$( \frac{3}{8}, \frac{3}{16}, \frac{11}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{16}^{1,3}$	24	$( \frac{3}{8}, \frac{5}{16}, \frac{13}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{16}^{7,6}$	25	$( \frac{3}{8}, \frac{7}{16}, \frac{15}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$3_{16}^{1,6}$	26	$( \frac{5}{8}, \frac{1}{16}, \frac{9}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{16}^{7,9}$	27	$( \frac{5}{8}, \frac{3}{16}, \frac{11}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{16}^{5,0}$	28	$( \frac{5}{8}, \frac{5}{16}, \frac{13}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$3_{16}^{3,3}$	29	$( \frac{5}{8}, \frac{7}{16}, \frac{15}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{16}^{5,3}$	30	$( \frac{7}{8}, \frac{1}{16}, \frac{9}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)
$3_{16}^{3,6}$	31	$( \frac{7}{8}, \frac{3}{16}, \frac{11}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{16}^{1,9}$	32	$( \frac{7}{8}, \frac{5}{16}, \frac{13}{16} )$,   (0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$)
$3_{16}^{7,0}$	33	$( \frac{7}{8}, \frac{7}{16}, \frac{15}{16} )$,   $\textrm{i}$(0, $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$4_{5,1}^{1}$	1	$( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} )$,   $(s_{5}^{2})^{-1}$($-\frac{5+\sqrt{5}}{10}$, $-\frac{\sqrt{15}}{5}$, $\frac{3-3\sqrt{5}}{2\sqrt{15}}$, $\frac{5-3\sqrt{5}}{10}$; $-\frac{5-3\sqrt{5}}{10}$, $\frac{5+\sqrt{5}}{10}$, $-\frac{3-3\sqrt{5}}{2\sqrt{15}}$; $\frac{5-3\sqrt{5}}{10}$, $-\frac{\sqrt{15}}{5}$; $\frac{5+\sqrt{5}}{10}$)
$4_{5,2}^{1}$	2	$( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} )$,   ($\frac{\sqrt{5}}{5}$, $-\frac{5-\sqrt{5}}{10}$, $-\frac{5+\sqrt{5}}{10}$, $\frac{\sqrt{5}}{5}$; $-\frac{\sqrt{5}}{5}$, $\frac{\sqrt{5}}{5}$, $\frac{5+\sqrt{5}}{10}$; $-\frac{\sqrt{5}}{5}$, $\frac{5-\sqrt{5}}{10}$; $\frac{\sqrt{5}}{5}$)
$4_{7}^{1}$	3	$( 0, \frac{1}{7}, \frac{2}{7}, \frac{4}{7} )$,   $\textrm{i}$($-\frac{\sqrt{7}}{7}$, $\frac{\sqrt{14}}{7}$, $\frac{\sqrt{14}}{7}$, $\frac{\sqrt{14}}{7}$; $-\frac{c_{7}^{2}}{\sqrt{7}}$, $-\frac{c_{7}^{1}}{\sqrt{7}}$, $-\frac{c_{7}^{3}}{\sqrt{7}}$; $-\frac{c_{7}^{3}}{\sqrt{7}}$, $-\frac{c_{7}^{2}}{\sqrt{7}}$; $-\frac{c_{7}^{1}}{\sqrt{7}}$)
$4_{7}^{3}$	4	$( 0, \frac{3}{7}, \frac{5}{7}, \frac{6}{7} )$,   $\textrm{i}$($\frac{\sqrt{7}}{7}$, $\frac{\sqrt{14}}{7}$, $\frac{\sqrt{14}}{7}$, $\frac{\sqrt{14}}{7}$; $\frac{c_{7}^{1}}{\sqrt{7}}$, $\frac{c_{7}^{2}}{\sqrt{7}}$, $\frac{c_{7}^{3}}{\sqrt{7}}$; $\frac{c_{7}^{3}}{\sqrt{7}}$, $\frac{c_{7}^{1}}{\sqrt{7}}$; $\frac{c_{7}^{2}}{\sqrt{7}}$)
$4_{8}^{1,0}$	5	$( \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8} )$,   $\textrm{i}$($\frac{\sqrt{2}}{4}$, $\frac{\sqrt{6}}{4}$, $\frac{\sqrt{6}}{4}$, $\frac{\sqrt{2}}{4}$; $\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{6}}{4}$; $-\frac{\sqrt{2}}{4}$, $\frac{\sqrt{6}}{4}$; $-\frac{\sqrt{2}}{4}$)
$4_{8}^{1,3}$	6	$( \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8} )$,   ($\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$, $\frac{\sqrt{6}}{4}$, $\frac{\sqrt{6}}{4}$; $-\frac{\sqrt{2}}{4}$, $\frac{\sqrt{6}}{4}$, $-\frac{\sqrt{6}}{4}$; $-\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{2}}{4}$; $\frac{\sqrt{2}}{4}$)
$4_{9,1}^{1,0}$	7	$( 0, \frac{1}{9}, \frac{4}{9}, \frac{7}{9} )$,   $\textrm{i}$(0, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$; $-\frac{1}{3}c_{36}^{1}$, $\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}c_{36}^{5}$; $\frac{1}{3}c_{36}^{5}$, $-\frac{1}{3}c_{36}^{1}$; $\frac{1}{3}c_{36}^{7}$)
$4_{9,1}^{2,0}$	8	$( 0, \frac{2}{9}, \frac{5}{9}, \frac{8}{9} )$,   $\textrm{i}$(0, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$; $-\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}c_{36}^{1}$, $-\frac{1}{3}c_{36}^{5}$; $-\frac{1}{3}c_{36}^{5}$, $-\frac{1}{3}c_{36}^{7}$; $\frac{1}{3}c_{36}^{1}$)
$4_{9,1}^{1,4}$	9	$( \frac{1}{3}, \frac{1}{9}, \frac{4}{9}, \frac{7}{9} )$,   $\textrm{i}$(0, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}c_{36}^{5}$, $-\frac{1}{3}c_{36}^{1}$; $-\frac{1}{3}c_{36}^{1}$, $\frac{1}{3}c_{36}^{7}$; $\frac{1}{3}c_{36}^{5}$)
$4_{9,1}^{2,4}$	10	$( \frac{1}{3}, \frac{2}{9}, \frac{5}{9}, \frac{8}{9} )$,   $\textrm{i}$(0, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{36}^{1}$, $-\frac{1}{3}c_{36}^{5}$, $-\frac{1}{3}c_{36}^{7}$; $-\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}c_{36}^{1}$; $-\frac{1}{3}c_{36}^{5}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$4_{9,1}^{1,8}$	11	$( \frac{2}{3}, \frac{1}{9}, \frac{4}{9}, \frac{7}{9} )$,   $\textrm{i}$(0, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{36}^{5}$, $-\frac{1}{3}c_{36}^{1}$, $\frac{1}{3}c_{36}^{7}$; $\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}c_{36}^{5}$; $-\frac{1}{3}c_{36}^{1}$)
$4_{9,1}^{2,8}$	12	$( \frac{2}{3}, \frac{2}{9}, \frac{5}{9}, \frac{8}{9} )$,   $\textrm{i}$(0, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$, $\frac{\sqrt{3}}{3}$; $-\frac{1}{3}c_{36}^{5}$, $-\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}c_{36}^{1}$; $\frac{1}{3}c_{36}^{1}$, $-\frac{1}{3}c_{36}^{5}$; $-\frac{1}{3}c_{36}^{7}$)
$4_{9,2}^{1,0}$	13	$( 0, \frac{1}{9}, \frac{4}{9}, \frac{7}{9} )$,   (0, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}c_{9}^{4}$, $\frac{1}{3}c_{9}^{1}$; $\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}c_{9}^{2}$; $\frac{1}{3}c_{9}^{4}$)
$4_{9,2}^{5,0}$	14	$( 0, \frac{2}{9}, \frac{5}{9}, \frac{8}{9} )$,   (0, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{9}^{4}$, $\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}c_{9}^{1}$; $\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}c_{9}^{4}$; $\frac{1}{3}c_{9}^{2}$)
$4_{9,2}^{1,4}$	15	$( \frac{1}{3}, \frac{1}{9}, \frac{4}{9}, \frac{7}{9} )$,   (0, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{9}^{4}$, $\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}c_{9}^{2}$; $\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}c_{9}^{4}$; $\frac{1}{3}c_{9}^{1}$)
$4_{9,2}^{5,4}$	16	$( \frac{1}{3}, \frac{2}{9}, \frac{5}{9}, \frac{8}{9} )$,   (0, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}c_{9}^{4}$; $\frac{1}{3}c_{9}^{4}$, $\frac{1}{3}c_{9}^{2}$; $\frac{1}{3}c_{9}^{1}$)
$4_{9,2}^{1,8}$	17	$( \frac{2}{3}, \frac{1}{9}, \frac{4}{9}, \frac{7}{9} )$,   (0, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}c_{9}^{4}$; $\frac{1}{3}c_{9}^{4}$, $\frac{1}{3}c_{9}^{1}$; $\frac{1}{3}c_{9}^{2}$)
$4_{9,2}^{5,8}$	18	$( \frac{2}{3}, \frac{2}{9}, \frac{5}{9}, \frac{8}{9} )$,   (0, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$, $-\frac{\sqrt{3}}{3}$; $\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}c_{9}^{4}$, $\frac{1}{3}c_{9}^{2}$; $\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}c_{9}^{1}$; $\frac{1}{3}c_{9}^{4}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$5_{5}^{1}$	1	$( 0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} )$,   ($-\frac{1}{5}$, $\frac{\sqrt{6}}{5}$, $\frac{\sqrt{6}}{5}$, $\frac{\sqrt{6}}{5}$, $\frac{\sqrt{6}}{5}$; $\frac{3-\sqrt{5}}{10}$, $-\frac{1+\sqrt{5}}{5}$, $-\frac{1-\sqrt{5}}{5}$, $\frac{3+\sqrt{5}}{10}$; $\frac{3+\sqrt{5}}{10}$, $\frac{3-\sqrt{5}}{10}$, $-\frac{1-\sqrt{5}}{5}$; $\frac{3+\sqrt{5}}{10}$, $-\frac{1+\sqrt{5}}{5}$; $\frac{3-\sqrt{5}}{10}$)
$5_{11}^{1}$	2	$( \frac{1}{11}, \frac{3}{11}, \frac{4}{11}, \frac{5}{11}, \frac{9}{11} )$,   ($-\frac{c_{44}^{3}}{\sqrt{11}}$, $-\frac{c_{44}^{7}}{\sqrt{11}}$, $-\frac{c_{44}^{5}}{\sqrt{11}}$, $-\frac{c_{44}^{1}}{\sqrt{11}}$, $-\frac{c_{44}^{9}}{\sqrt{11}}$; $\frac{c_{44}^{9}}{\sqrt{11}}$, $-\frac{c_{44}^{3}}{\sqrt{11}}$, $\frac{c_{44}^{5}}{\sqrt{11}}$, $\frac{c_{44}^{1}}{\sqrt{11}}$; $\frac{c_{44}^{1}}{\sqrt{11}}$, $\frac{c_{44}^{9}}{\sqrt{11}}$, $\frac{c_{44}^{7}}{\sqrt{11}}$; $\frac{c_{44}^{7}}{\sqrt{11}}$, $-\frac{c_{44}^{3}}{\sqrt{11}}$; $\frac{c_{44}^{5}}{\sqrt{11}}$)
$5_{11}^{2}$	3	$( \frac{2}{11}, \frac{6}{11}, \frac{7}{11}, \frac{8}{11}, \frac{10}{11} )$,   ($\frac{c_{44}^{5}}{\sqrt{11}}$, $\frac{c_{44}^{3}}{\sqrt{11}}$, $-\frac{c_{44}^{7}}{\sqrt{11}}$, $-\frac{c_{44}^{1}}{\sqrt{11}}$, $-\frac{c_{44}^{9}}{\sqrt{11}}$; $\frac{c_{44}^{7}}{\sqrt{11}}$, $\frac{c_{44}^{9}}{\sqrt{11}}$, $\frac{c_{44}^{5}}{\sqrt{11}}$, $\frac{c_{44}^{1}}{\sqrt{11}}$; $\frac{c_{44}^{1}}{\sqrt{11}}$, $-\frac{c_{44}^{3}}{\sqrt{11}}$, $\frac{c_{44}^{5}}{\sqrt{11}}$; $\frac{c_{44}^{9}}{\sqrt{11}}$, $\frac{c_{44}^{7}}{\sqrt{11}}$; $-\frac{c_{44}^{3}}{\sqrt{11}}$)

$d_{l,k}^{a,m}$	#	$\rho (\mathfrak {t})$, $\rho (\mathfrak {s})$
$6_{5}^{1}$	1	$( 0, 0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} )$,   $(s_{5}^{2})^{-1}$($\frac{\sqrt{5}}{5}$, $-\frac{5-\sqrt{5}}{10}$, $-\frac{\sqrt{10}}{5}$, $-\frac{5+\sqrt{5}}{10}$, $\frac{5-3\sqrt{5}}{10}$, $\frac{1-\sqrt{5}}{\sqrt{10}}$; $-\frac{\sqrt{5}}{5}$, $\frac{-1+\sqrt{5}}{\sqrt{10}}$, $\frac{5-3\sqrt{5}}{10}$, $\frac{5+\sqrt{5}}{10}$, $-\frac{\sqrt{10}}{5}$; $\frac{5-\sqrt{5}}{10}$, $\frac{1-\sqrt{5}}{\sqrt{10}}$, $-\frac{\sqrt{10}}{5}$, $-\frac{\sqrt{5}}{5}$; $-\frac{\sqrt{5}}{5}$, $\frac{5-\sqrt{5}}{10}$, $\frac{\sqrt{10}}{5}$; $\frac{\sqrt{5}}{5}$, $\frac{1-\sqrt{5}}{\sqrt{10}}$; $-\frac{5-\sqrt{5}}{10}$)
$6_{7,1}^{1}$	2	$( \frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7} )$,   $\textrm{i}$($\frac{1}{7}c^{2}_{56} -\frac{1}{7} c^{3}_{56} +\frac{1}{7}c^{11}_{56}$, $\frac{1}{7}c^{5}_{56} +\frac{1}{7}c^{6}_{56} +\frac{1}{7}c^{9}_{56}$, $\frac{1}{7}c^{3}_{112} -\frac{1}{7} c^{9}_{112} +\frac{1}{7}c^{11}_{112} +\frac{1}{7}c^{23}_{112}$, $\frac{2}{7}c^{1}_{56} -\frac{1}{7} c^{3}_{56} -\frac{1}{7} c^{5}_{56} +\frac{1}{7}c^{7}_{56} +\frac{1}{7}c^{9}_{56} -\frac{1}{7} c^{10}_{56} -\frac{1}{7} c^{11}_{56}$, $\frac{1}{7}c^{1}_{112} +\frac{1}{7}c^{3}_{112} -\frac{1}{7} c^{5}_{112} -\frac{1}{7} c^{7}_{112} -\frac{1}{7} c^{9}_{112} -\frac{1}{7} c^{11}_{112} +\frac{2}{7}c^{13}_{112} +\frac{1}{7}c^{15}_{112} +\frac{2}{7}c^{17}_{112} +\frac{1}{7}c^{19}_{112} -\frac{1}{7} c^{21}_{112} -\frac{1}{7} c^{23}_{112}$, $\frac{1}{7}c^{1}_{112} +\frac{1}{7}c^{5}_{112} -\frac{1}{7} c^{15}_{112} +\frac{1}{7}c^{19}_{112}$; $\frac{2}{7}c^{1}_{56} -\frac{1}{7} c^{3}_{56} -\frac{1}{7} c^{5}_{56} +\frac{1}{7}c^{7}_{56} +\frac{1}{7}c^{9}_{56} -\frac{1}{7} c^{10}_{56} -\frac{1}{7} c^{11}_{56}$, $\frac{1}{7}c^{1}_{112} +\frac{1}{7}c^{5}_{112} -\frac{1}{7} c^{15}_{112} +\frac{1}{7}c^{19}_{112}$, $\frac{1}{7}c^{2}_{56} -\frac{1}{7} c^{3}_{56} +\frac{1}{7}c^{11}_{56}$, $-\frac{1}{7} c^{3}_{112} +\frac{1}{7}c^{9}_{112} -\frac{1}{7} c^{11}_{112} -\frac{1}{7} c^{23}_{112}$, $-\frac{1}{7} c^{1}_{112} -\frac{1}{7} c^{3}_{112} +\frac{1}{7}c^{5}_{112} +\frac{1}{7}c^{7}_{112} +\frac{1}{7}c^{9}_{112} +\frac{1}{7}c^{11}_{112} -\frac{2}{7} c^{13}_{112} -\frac{1}{7} c^{15}_{112} -\frac{2}{7} c^{17}_{112} -\frac{1}{7} c^{19}_{112} +\frac{1}{7}c^{21}_{112} +\frac{1}{7}c^{23}_{112}$; $-\frac{1}{7} c^{5}_{56} -\frac{1}{7} c^{6}_{56} -\frac{1}{7} c^{9}_{56}$, $-\frac{1}{7} c^{1}_{112} -\frac{1}{7} c^{3}_{112} +\frac{1}{7}c^{5}_{112} +\frac{1}{7}c^{7}_{112} +\frac{1}{7}c^{9}_{112} +\frac{1}{7}c^{11}_{112} -\frac{2}{7} c^{13}_{112} -\frac{1}{7} c^{15}_{112} -\frac{2}{7} c^{17}_{112} -\frac{1}{7} c^{19}_{112} +\frac{1}{7}c^{21}_{112} +\frac{1}{7}c^{23}_{112}$, $\frac{1}{7}c^{2}_{56} -\frac{1}{7} c^{3}_{56} +\frac{1}{7}c^{11}_{56}$, $-\frac{2}{7} c^{1}_{56} +\frac{1}{7}c^{3}_{56} +\frac{1}{7}c^{5}_{56} -\frac{1}{7} c^{7}_{56} -\frac{1}{7} c^{9}_{56} +\frac{1}{7}c^{10}_{56} +\frac{1}{7}c^{11}_{56}$; $\frac{1}{7}c^{5}_{56} +\frac{1}{7}c^{6}_{56} +\frac{1}{7}c^{9}_{56}$, $-\frac{1}{7} c^{1}_{112} -\frac{1}{7} c^{5}_{112} +\frac{1}{7}c^{15}_{112} -\frac{1}{7} c^{19}_{112}$, $\frac{1}{7}c^{3}_{112} -\frac{1}{7} c^{9}_{112} +\frac{1}{7}c^{11}_{112} +\frac{1}{7}c^{23}_{112}$; $-\frac{2}{7} c^{1}_{56} +\frac{1}{7}c^{3}_{56} +\frac{1}{7}c^{5}_{56} -\frac{1}{7} c^{7}_{56} -\frac{1}{7} c^{9}_{56} +\frac{1}{7}c^{10}_{56} +\frac{1}{7}c^{11}_{56}$, $\frac{1}{7}c^{5}_{56} +\frac{1}{7}c^{6}_{56} +\frac{1}{7}c^{9}_{56}$; $-\frac{1}{7} c^{2}_{56} +\frac{1}{7}c^{3}_{56} -\frac{1}{7} c^{11}_{56}$)
$6_{7,2}^{1}$	3	$( \frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7} )$,   ($\frac{2}{7}-\frac{1}{7} c^{2}_{7}$, $-\frac{2}{7}+\frac{1}{7}c^{1}_{7}$, $\frac{1}{7}c^{3}_{56} +\frac{2}{7}c^{5}_{56} -\frac{1}{7} c^{7}_{56} -\frac{2}{7} c^{9}_{56} +\frac{1}{7}c^{11}_{56}$, $-\frac{3}{7}-\frac{1}{7} c^{1}_{7} -\frac{1}{7} c^{2}_{7}$, $-\frac{1}{7} c^{3}_{56} +\frac{1}{7}c^{5}_{56} -\frac{1}{7} c^{9}_{56} -\frac{1}{7} c^{11}_{56}$, $-\frac{2}{7} c^{3}_{56} -\frac{1}{7} c^{5}_{56} +\frac{1}{7}c^{7}_{56} +\frac{1}{7}c^{9}_{56} -\frac{2}{7} c^{11}_{56}$; $\frac{3}{7}+\frac{1}{7}c^{1}_{7} +\frac{1}{7}c^{2}_{7}$, $\frac{2}{7}c^{3}_{56} +\frac{1}{7}c^{5}_{56} -\frac{1}{7} c^{7}_{56} -\frac{1}{7} c^{9}_{56} +\frac{2}{7}c^{11}_{56}$, $\frac{2}{7}-\frac{1}{7} c^{2}_{7}$, $\frac{1}{7}c^{3}_{56} +\frac{2}{7}c^{5}_{56} -\frac{1}{7} c^{7}_{56} -\frac{2}{7} c^{9}_{56} +\frac{1}{7}c^{11}_{56}$, $-\frac{1}{7} c^{3}_{56} +\frac{1}{7}c^{5}_{56} -\frac{1}{7} c^{9}_{56} -\frac{1}{7} c^{11}_{56}$; $\frac{2}{7}-\frac{1}{7} c^{1}_{7}$, $-\frac{1}{7} c^{3}_{56} +\frac{1}{7}c^{5}_{56} -\frac{1}{7} c^{9}_{56} -\frac{1}{7} c^{11}_{56}$, $-\frac{2}{7}+\frac{1}{7}c^{2}_{7}$, $\frac{3}{7}+\frac{1}{7}c^{1}_{7} +\frac{1}{7}c^{2}_{7}$; $\frac{2}{7}-\frac{1}{7} c^{1}_{7}$, $-\frac{2}{7} c^{3}_{56} -\frac{1}{7} c^{5}_{56} +\frac{1}{7}c^{7}_{56} +\frac{1}{7}c^{9}_{56} -\frac{2}{7} c^{11}_{56}$, $-\frac{1}{7} c^{3}_{56} -\frac{2}{7} c^{5}_{56} +\frac{1}{7}c^{7}_{56} +\frac{2}{7}c^{9}_{56} -\frac{1}{7} c^{11}_{56}$; $\frac{3}{7}+\frac{1}{7}c^{1}_{7} +\frac{1}{7}c^{2}_{7}$, $-\frac{2}{7}+\frac{1}{7}c^{1}_{7}$; $\frac{2}{7}-\frac{1}{7} c^{2}_{7}$)
$6_{8,1}^{1,0}$	4	$( 0, \frac{1}{2}, \frac{1}{8}, \frac{3}{8}, \frac{5}{8}, \frac{7}{8} )$,   (0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; 0, $-\frac{1}{2}$, 0, $\frac{1}{2}$; 0, $\frac{1}{2}$, 0; 0, $-\frac{1}{2}$; 0)
$6_{8,2}^{1,0}$	5	$( 0, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{1}{8}, \frac{3}{8} )$,   $\textrm{i}$($-\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$, $\frac{1}{2}$, $\frac{1}{2}$; $-\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{2}}{4}$, $\frac{1}{2}$, $\frac{1}{2}$; $\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$, $-\frac{1}{2}$, $\frac{1}{2}$; $\frac{\sqrt{2}}{4}$, $\frac{1}{2}$, $-\frac{1}{2}$; 0, 0; 0)

$6_{9,1}^{1,0}$	6	$( \frac{1}{9}, \frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}, \frac{8}{9} )$,   $\textrm{i}$($-\frac{1}{3}$, $\frac{1}{3}c_{72}^{7}$, $\frac{1}{3}$, $-\frac{1}{3}c_{72}^{17}$, $\frac{1}{3}$, $-\frac{1}{3}c_{72}^{5}$; $\frac{1}{3}$, $\frac{1}{3}c_{72}^{5}$, $-\frac{1}{3}$, $-\frac{1}{3}c_{72}^{17}$, $\frac{1}{3}$; $-\frac{1}{3}$, $\frac{1}{3}c_{72}^{7}$, $-\frac{1}{3}$, $-\frac{1}{3}c_{72}^{17}$; $\frac{1}{3}$, $-\frac{1}{3}c_{72}^{5}$, $-\frac{1}{3}$; $-\frac{1}{3}$, $-\frac{1}{3}c_{72}^{7}$; $\frac{1}{3}$)
$6_{9,2}^{1,0}$	7	$( \frac{1}{9}, \frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}, \frac{8}{9} )$,   ($-\frac{1}{3}$, $\frac{1}{3}c_{36}^{1}$, $\frac{1}{3}$, $\frac{1}{3}c_{36}^{5}$, $\frac{1}{3}$, $-\frac{1}{3}c_{36}^{7}$; $-\frac{1}{3}$, $\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}$, $\frac{1}{3}c_{36}^{5}$, $-\frac{1}{3}$; $-\frac{1}{3}$, $\frac{1}{3}c_{36}^{1}$, $-\frac{1}{3}$, $\frac{1}{3}c_{36}^{5}$; $-\frac{1}{3}$, $-\frac{1}{3}c_{36}^{7}$, $\frac{1}{3}$; $-\frac{1}{3}$, $-\frac{1}{3}c_{36}^{1}$; $-\frac{1}{3}$)
$6_{9,3}^{1,0}$	8	$( \frac{1}{9}, \frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}, \frac{8}{9} )$,   ($\frac{1}{3}$, $\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}$, $-\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}$, $\frac{1}{3}c_{9}^{4}$; $\frac{1}{3}$, $\frac{1}{3}c_{9}^{4}$, $-\frac{1}{3}$, $\frac{1}{3}c_{9}^{1}$, $\frac{1}{3}$; $\frac{1}{3}$, $-\frac{1}{3}c_{9}^{2}$, $\frac{1}{3}$, $\frac{1}{3}c_{9}^{1}$; $\frac{1}{3}$, $-\frac{1}{3}c_{9}^{4}$, $-\frac{1}{3}$; $\frac{1}{3}$, $\frac{1}{3}c_{9}^{2}$; $\frac{1}{3}$)
$6_{11}^{1}$	9	$( 0, \frac{1}{11}, \frac{3}{11}, \frac{4}{11}, \frac{5}{11}, \frac{9}{11} )$,   $\textrm{i}$($-\frac{\sqrt{11}}{11}$, $\frac{\sqrt{22}}{11}$, $\frac{\sqrt{22}}{11}$, $\frac{\sqrt{22}}{11}$, $\frac{\sqrt{22}}{11}$, $\frac{\sqrt{22}}{11}$; $-\frac{c_{11}^{2}}{\sqrt{11}}$, $-\frac{c_{11}^{1}}{\sqrt{11}}$, $-\frac{c_{11}^{4}}{\sqrt{11}}$, $-\frac{c_{11}^{3}}{\sqrt{11}}$, $-\frac{c_{11}^{5}}{\sqrt{11}}$; $-\frac{c_{11}^{5}}{\sqrt{11}}$, $-\frac{c_{11}^{2}}{\sqrt{11}}$, $-\frac{c_{11}^{4}}{\sqrt{11}}$, $-\frac{c_{11}^{3}}{\sqrt{11}}$; $-\frac{c_{11}^{3}}{\sqrt{11}}$, $-\frac{c_{11}^{5}}{\sqrt{11}}$, $-\frac{c_{11}^{1}}{\sqrt{11}}$; $-\frac{c_{11}^{1}}{\sqrt{11}}$, $-\frac{c_{11}^{2}}{\sqrt{11}}$; $-\frac{c_{11}^{4}}{\sqrt{11}}$)
$6_{13}^{1}$	10	$( \frac{1}{13}, \frac{3}{13}, \frac{4}{13}, \frac{9}{13}, \frac{10}{13}, \frac{12}{13} )$,   $\textrm{i}$($-\frac{c_{52}^{5}}{\sqrt{13}}$, $\frac{c_{52}^{7}}{\sqrt{13}}$, $\frac{c_{52}^{3}}{\sqrt{13}}$, $\frac{c_{52}^{11}}{\sqrt{13}}$, $\frac{c_{52}^{9}}{\sqrt{13}}$, $-\frac{c_{52}^{1}}{\sqrt{13}}$; $-\frac{c_{52}^{11}}{\sqrt{13}}$, $\frac{c_{52}^{1}}{\sqrt{13}}$, $-\frac{c_{52}^{5}}{\sqrt{13}}$, $\frac{c_{52}^{3}}{\sqrt{13}}$, $\frac{c_{52}^{9}}{\sqrt{13}}$; $\frac{c_{52}^{7}}{\sqrt{13}}$, $\frac{c_{52}^{9}}{\sqrt{13}}$, $-\frac{c_{52}^{5}}{\sqrt{13}}$, $\frac{c_{52}^{11}}{\sqrt{13}}$; $-\frac{c_{52}^{7}}{\sqrt{13}}$, $-\frac{c_{52}^{1}}{\sqrt{13}}$, $-\frac{c_{52}^{3}}{\sqrt{13}}$; $\frac{c_{52}^{11}}{\sqrt{13}}$, $-\frac{c_{52}^{7}}{\sqrt{13}}$; $\frac{c_{52}^{5}}{\sqrt{13}}$)
$6_{16,1}^{1,0}$	11	$( 0, \frac{1}{4}, \frac{1}{16}, \frac{5}{16}, \frac{9}{16}, \frac{13}{16} )$,   $\textrm{i}$(0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; $-\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$; $\frac{\sqrt{2}}{4}$, $\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{2}}{4}$; $-\frac{\sqrt{2}}{4}$, $-\frac{\sqrt{2}}{4}$; $\frac{\sqrt{2}}{4}$)
$6_{16,2}^{1,0}$	12	$( 0, \frac{1}{2}, \frac{1}{16}, \frac{3}{16}, \frac{9}{16}, \frac{11}{16} )$,   $\textrm{i}$(0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; 0, $-\frac{1}{2}$, 0, $\frac{1}{2}$; 0, $\frac{1}{2}$, 0; 0, $-\frac{1}{2}$; 0)
$6_{16,3}^{1,0}$	13	$( 0, \frac{1}{2}, \frac{1}{16}, \frac{7}{16}, \frac{9}{16}, \frac{15}{16} )$,   (0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; 0, $-\frac{1}{2}$, 0, $\frac{1}{2}$; 0, $\frac{1}{2}$, 0; 0, $-\frac{1}{2}$; 0)
$6_{16,4}^{1,0}$	14	$( \frac{1}{8}, \frac{5}{8}, \frac{1}{16}, \frac{5}{16}, \frac{9}{16}, \frac{13}{16} )$,   (0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; 0, $-\frac{1}{2}$, 0, $\frac{1}{2}$; 0, $\frac{1}{2}$, 0; 0, $-\frac{1}{2}$; 0)
$6_{32,1}^{1,0}$	15	$( 0, \frac{1}{8}, \frac{3}{32}, \frac{11}{32}, \frac{19}{32}, \frac{27}{32} )$,   $\textrm{i}$(0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; $-\frac{1}{4}c_{16}^{1}$, $-\frac{1}{4}c_{16}^{3}$, $\frac{1}{4}c_{16}^{1}$, $\frac{1}{4}c_{16}^{3}$; $\frac{1}{4}c_{16}^{1}$, $\frac{1}{4}c_{16}^{3}$, $-\frac{1}{4}c_{16}^{1}$; $-\frac{1}{4}c_{16}^{1}$, $-\frac{1}{4}c_{16}^{3}$; $\frac{1}{4}c_{16}^{1}$)
$6_{32,2}^{1,0}$	16	$( 0, \frac{1}{8}, \frac{7}{32}, \frac{15}{32}, \frac{23}{32}, \frac{31}{32} )$,   (0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; $-\frac{1}{4}c_{16}^{1}$, $-\frac{1}{4}c_{16}^{3}$, $\frac{1}{4}c_{16}^{1}$, $\frac{1}{4}c_{16}^{3}$; $\frac{1}{4}c_{16}^{1}$, $\frac{1}{4}c_{16}^{3}$, $-\frac{1}{4}c_{16}^{1}$; $-\frac{1}{4}c_{16}^{1}$, $-\frac{1}{4}c_{16}^{3}$; $\frac{1}{4}c_{16}^{1}$)

A List of All Candidate ${\text {SL}}_2({\mathbb {Z}})$ Representations of MTCs

We will follow the strategy outlined in Sect. 3.4. We first try to obtain a list that includes all ${\text {SL}}_2({\mathbb {Z}})$ representations associated with MTCs. Certainly, one such list is the list of all ${\text {SL}}_2({\mathbb {Z}})$ representations of finite levels. But such a list is very inefficient since most representations in the list are not associated with MTCs. So in this section we collect the conditions that a representation coming from a MTC must satisfy, to obtain a shorter list.

1.1 The conditions on ${\text {SL}}_2({\mathbb {Z}})$ representations

Some of the conditions on ${\text {SL}}_2({\mathbb {Z}})$ representations are obtained from the necessary conditions on modular data Propositions B.1 and 3.7, and others are discussed in the main text of this paper. Let us first translate the conditions on the (S, T) matrices to condition on an MD representations $\rho _\alpha$:

Proposition B.1

Given a modular data S, T of rank r, let $\rho _\alpha$ be any one of its 12 MD representations. Then $\rho _\alpha$ has the following properties:

• (1) $\rho _\alpha$ is an ${\text {SL}}_2({\mathbb {Z}})$ representation of level ${\text {ord}}(\rho _\alpha (\mathfrak {t}))$, and ${\text {ord}}(T) \mid {\text {ord}}(\rho _\alpha (\mathfrak {t})) \mid 12 {\text {ord}}(T)$ .

• (2) The conductor of the elements of $\rho _\alpha (\mathfrak {s})$ divides ${\text {ord}}(\rho _\alpha (\mathfrak {t}))$.

• (3) If $\rho _\alpha$ is a direct sum of two ${\text {SL}}_2({\mathbb {Z}})$ representations

  $$\begin{aligned} \rho _\alpha \cong \rho \oplus \rho ', \end{aligned}$$

  (B.1)

  then the eigenvalues of $\rho (\mathfrak {t})$ and $\rho '(\mathfrak {t})$ must overlap. This implies that if $\rho _\alpha = \rho \oplus \chi _1 \oplus \dots \oplus \chi _\ell$ for some 1-dimensional representations $\chi _1, \dots , \chi _\ell$, then $\chi _1, \cdots \chi _\ell$ are the same 1-dimensional representation.

• (4) Suppose that $\rho _\alpha \cong \rho \oplus \ell \chi$ for an irreducible representation $\rho$ with non-degenerate $\rho (\mathfrak {t})$, and an 1-dimensional representation $\chi$. If $\ell \ne 2\dim (\rho ) -1$ or $\ell > 1$, then $(\rho (\mathfrak {s}) \chi (\mathfrak {s})^{-1})^2 = {\text {id}}$.

• (5) $\rho _\alpha$ satisfies

  $$\begin{aligned} \rho _\alpha \not \cong n \rho \end{aligned}$$

  (B.2)

  for any integer $n >1$ and any representation $\rho$ such that $\rho (\mathfrak {t})$ is non-degenerate.

• (6) If $\rho _\alpha (s)^2=\pm {\text {id}}$ (i.e. if the modular data or MTC is self dual), ${\text {pord}}(\rho _\alpha (\mathfrak {t}))$ is a prime and satisfies ${\text {pord}}(\rho _\alpha (\mathfrak {t})) = 1$ mod 4, then the representation $\rho _\alpha$ cannot be a direct sum of a d-dimensional irreducible ${\text {SL}}_2({\mathbb {Z}})$ representation and two or more 1-dimensional ${\text {SL}}_2({\mathbb {Z}})$ representations with $d=(p+1)/2$.

• (7) Let $3< p < q$ be prime such that $pq \equiv 3 \mod 4$ and ${\text {pord}}(\rho _\alpha (\mathfrak {t}))=pq$, then the rank $r \ne \frac{p+q}{2}+1$. Moreover, if $p > 5$, rank $r > \frac{p+q}{2}+1$.

• (8) The number of self dual objects is greater than 0. Thus

  $$\begin{aligned} {\text {Tr}}(\rho _\alpha (\mathfrak {s})^2) \ne 0 . \end{aligned}$$

  (B.3)

  Since ${\text {Tr}}(\rho _\alpha (\mathfrak {s})^2) \ne 0$, let us introduce

  $$\begin{aligned} C = \frac{{\text {Tr}}(\rho _\alpha (\mathfrak {s})^2)}{|{\text {Tr}}(\rho _\alpha (\mathfrak {s})^2)|} \rho _\alpha (\mathfrak {s})^2. \end{aligned}$$

  (B.4)

  The above C is the charge conjugation operator of MTC, i.e. C is a permutation matrix of order 2. In particular, ${\text {Tr}}(C)$ is the number of self dual objects. Also, for each eigenvalue $\tilde{\theta }$ of $\rho _\alpha (\mathfrak {t})$,

  $$\begin{aligned} {\text {Tr}}_{\tilde{\theta }}(C) \geqslant 0, \end{aligned}$$

  (B.5)

  where ${\text {Tr}}_{\tilde{\theta }}$ is the trace in the degenerate subspace of $\rho _\alpha (\mathfrak {t})$ with eigenvalue $\tilde{\theta }$.

• (9) For any Galois conjugation $\sigma$ in ${\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\rho _\alpha (\mathfrak {t}))})$, there is a permutation of the indices, $i \rightarrow {\hat{\sigma }}(i)$, and $\epsilon _\sigma (i)\in \{1,-1\}$, such that

  $$\begin{aligned} \sigma \big (\rho _\alpha (\mathfrak {s})_{i,j}\big )&= \epsilon _\sigma (i)\rho _\alpha (\mathfrak {s})_{{\hat{\sigma }} (i),j} = \rho _\alpha (\mathfrak {s})_{i,{\hat{\sigma }} (j)}\epsilon _\sigma (j) \end{aligned}$$

  (B.6)
  $$\begin{aligned} \sigma ^2 \big (\rho _\alpha (\mathfrak {t})_{i,i}\big )&= \rho _\alpha (\mathfrak {t})_{{\hat{\sigma }} (i),\hat{\sigma }(i)}, \end{aligned}$$

  (B.7)

  for all i, j.

• (10) By [11, Theorem II], $D_{\rho _\alpha }(\sigma )$ defined in (3.6) must be a signed permutation

  $$\begin{aligned} (D_{\rho _\alpha }(\sigma ))_{i,j} = \epsilon _\sigma (i) \delta _{{\hat{\sigma }}(i),j}. \end{aligned}$$

  and satisfies

  $$\begin{aligned} \sigma (\rho _\alpha (\mathfrak {s}))&= D_{\rho _\alpha }(\sigma ) \rho _\alpha (\mathfrak {s}) =\rho _\alpha (\mathfrak {s})D_{\rho _\alpha }^\top (\sigma ), \nonumber \\ \sigma ^2(\rho _\alpha (\mathfrak {t}))&= D_{\rho _\alpha }(\sigma ) \rho _\alpha (\mathfrak {t}) D_{\rho _\alpha }^\top (\sigma ) \end{aligned}$$

  (B.8)

• (11) There exists a u such that $\rho _\alpha (\mathfrak {s})_{uu} \ne 0$ and

  $$\begin{aligned}&\rho _\alpha (\mathfrak {s})_{ui} \ne 0 \in {\mathbb {R}}, \ \ \ \ \frac{ \rho _\alpha (\mathfrak {s})_{ij} }{ \rho _\alpha (\mathfrak {s})_{uu} },\ \frac{ \rho _\alpha (\mathfrak {s})_{ij} }{ \rho _\alpha (\mathfrak {s})_{uj} } \in {\mathbb {O}}_{{\text {ord}}(T)}, \ \ \ \ \frac{ \rho _\alpha (\mathfrak {s})_{ij} }{ \rho _\alpha (\mathfrak {s})_{i'j'} } \in {\mathbb {Q}}_{{\text {ord}}(T)}, \nonumber \\&N^{ij}_k = \sum _{l=0}^{r-1} \frac{ \rho _\alpha (\mathfrak {s})_{li} \rho _\alpha (\mathfrak {s})_{lj} \rho _\alpha (\mathfrak {s}^{-1})_{lk}}{ \rho _\alpha (\mathfrak {s})_{lu} } \in {\mathbb {N}}. \nonumber \\&\forall \ i,j,k = 0,1,\ldots ,r-1. \end{aligned}$$

  (B.9)

  (u corresponds the unit object of MTC).

• (12) Let $n \in {\mathbb {N}}_+$. The $n^\text {th}$ Frobenius-Schur indicator of the i-th simple object

  $$\begin{aligned} \nu _n(i)&= \sum _{j, k=0}^{r-1} N_i^{jk} \rho _\alpha (\mathfrak {s})_{ju}\theta _j^n [\rho _\alpha (\mathfrak {s})_{ku}\theta _k^n]^* =\sum _{j, k=0}^{r-1} N_i^{jk} \rho _\alpha (\mathfrak {t}^n\mathfrak {s})_{ju} \rho _\alpha (\mathfrak {t}^{-n}\mathfrak {s}^{-1})_{ku} \nonumber \\&=\sum _{j,k,l=0}^{r-1} \frac{ \rho _\alpha (\mathfrak {s})_{lj} \rho _\alpha (\mathfrak {s})_{lk} \rho ^*_\alpha (\mathfrak {s})_{li}}{ \rho _\alpha (\mathfrak {s})_{lu} } \rho _\alpha (\mathfrak {t}^n\mathfrak {s})_{ju} \rho _\alpha (\mathfrak {t}^{-n}\mathfrak {s}^{-1})_{ku} \nonumber \\&=\sum _{l=0}^{r-1} \frac{ \rho _\alpha (\mathfrak {s}\mathfrak {t}^n\mathfrak {s})_{lu} \rho _\alpha (\mathfrak {s}\mathfrak {t}^{-n}\mathfrak {s}^{-1})_{lu} \rho _\alpha (\mathfrak {s}^{-1})_{li}}{ \rho _\alpha (\mathfrak {s})_{lu} } \end{aligned}$$

  (B.10)

  is a cyclotomic integer whose conductor divides n and ${\text {ord}}(T)$. The 1st Frobenius-Schur indicator satisfies $\nu _1(i)=\delta _{iu}$ while the 2nd Frobenius-Schur indicator $\nu _2(i)$ satisfies $\nu _2(i)=\pm \rho _\alpha (\mathfrak {s}^2)_{ii}$ (see [3, 24, 33]).

• (13) If we further assume the modular data or the MTC to be non-integral, then ${\text {pord}}(\tilde{\rho }_\alpha (\mathfrak {t})) = {\text {ord}}(T) \notin \{2,3,4,6\}$. In particular, ${\text {ord}}(\rho _\alpha (\mathfrak {t})) \notin \{2,3,4,6\}$.

In Sect. 3.1 and “Appendix A”, we have explicitly constructed all irreducible unitary representations of ${\text {SL}}_2({\mathbb {Z}})$ (up to unitary equivalence). However, this only gives the ${\text {SL}}_2({\mathbb {Z}})$ representations in some arbitrary basis, not in the basis yielding MD representations (i.e. satisfying (3.7)). We can improve the situation by choosing a basis to make $\rho (\mathfrak {t})$ diagonal and $\rho (\mathfrak {s})$ symmetric. Since we are going to use several types of bases, let us define these choices:

Definition B.2

An unitary ${\text {SL}}_2({\mathbb {Z}})$ representations $\tilde{\rho }$ is called a general ${\text {SL}}_2({\mathbb {Z}})$ matrix representations if $\tilde{\rho }(\mathfrak {t})$ is diagonal.⁷ A general ${\text {SL}}_2({\mathbb {Z}})$ matrix representation $\tilde{\rho }$ is called symmetric if $\tilde{\rho }(\mathfrak {s})$ is symmetric. An general ${\text {SL}}_2({\mathbb {Z}})$ matrix representation $\tilde{\rho }$ is called irrep-sum if $\tilde{\rho }(\mathfrak {s}),\tilde{\rho }(\mathfrak {t})$ are matrix-direct sum of irreducible ${\text {SL}}_2({\mathbb {Z}})$ representations. An ${\text {SL}}_2({\mathbb {Z}})$ matrix representations $\tilde{\rho }$ is called an ${\text {SL}}_2({\mathbb {Z}})$ representation of modular data S, T, if $\tilde{\rho }$ is unitary equivalent to an MD representation of the modular data, i.e.,

$$\begin{aligned} \tilde{\rho }(\mathfrak {s}) = \textrm{e}^{-2\pi \textrm{i}\frac{\alpha }{4}}\frac{1}{D}\, US U^\dag , \ \ \ \ \tilde{\rho }(\mathfrak {t}) = UTU^\dag \textrm{e}^{2\pi \textrm{i}(\frac{-c}{24} + \frac{\alpha }{12})}, \end{aligned}$$

(B.11)

for some unitary matrix U and $\alpha \in {\mathbb {Z}}_{12}$, where c is the central charge.⁸

Through our explicit construction, we observe that all irreducible unitary representations of ${\text {SL}}_2({\mathbb {Z}})$ are unitarily equivalent to symmetric matrix representations of ${\text {SL}}_2({\mathbb {Z}})$, at least for dimension equal or less than 12.

We note that different choices of orthogonal basis give rise to different matrix representations of ${\text {SL}}_2({\mathbb {Z}})$. The modular data S, T is obtained from some particular choices of the basis. Some properties on the MD representations of a modular data do not depend on the choices of basis in the eigenspaces of $\tilde{\rho }(\mathfrak {t})$ (induced by the block-diagonal unitary transformation U in (B.11) that leaves $\tilde{\rho }(\mathfrak {t})$ invariant). Those properties remain valid for any general ${\text {SL}}_2({\mathbb {Z}})$ representations $\tilde{\rho }$ of the modular data. In the following, we collect the basis-independent conditions on the ${\text {SL}}_2({\mathbb {Z}})$ matrix representations of modular data. Those conditions have been discussed in the main text.

Proposition B.3

Let $\tilde{\rho }$ be a general ${\text {SL}}_2({\mathbb {Z}})$ matrix representations of a modular data or a MTC. Then $\tilde{\rho }$ must satisfy the following conditions:

• (1) If $\tilde{\rho }$ is a direct sum of two ${\text {SL}}_2({\mathbb {Z}})$ representations

  $$\begin{aligned} \tilde{\rho }\cong \rho \oplus \rho ', \end{aligned}$$

  (B.12)

  then the diagonals entries of $\rho (\mathfrak {t})$ and $\rho '(\mathfrak {t})$ must overlap.

• (2) Suppose that $\tilde{\rho }\cong \rho \oplus \ell \chi$ for an irreducible representation $\rho$ with $\rho (\mathfrak {t})$ non-degenerate, and a character $\chi$. If $\ell \ne 1$ and $\ell \ne 2\dim (\rho ) -1$, then $(\rho (\mathfrak {s}) \chi (\mathfrak {s})^{-1})^2 = {\text {id}}$.

• (3) If $\tilde{\rho }(\mathfrak {s})^2=\pm {\text {id}}$, and ${\text {pord}}(\tilde{\rho }(\mathfrak {t})) = 1$ mod 4 is a prime, then the representation $\tilde{\rho }$ cannot be a direct sum of a d-dimensional irreducible ${\text {SL}}_2({\mathbb {Z}})$ representation and two or more 1-dimensional ${\text {SL}}_2({\mathbb {Z}})$ representations with $d=({\text {pord}}(\tilde{\rho }(\mathfrak {t})) +1)/2$.

• (4) $\tilde{\rho }$ satisfies

  $$\begin{aligned} \tilde{\rho }\not \cong n \rho \end{aligned}$$

  (B.13)

  for any integer $n >1$ and any representation $\rho$ such that $\rho (\mathfrak {t})$ is non-degenerate.

• (5) Let $3< p < q$ be prime such that $pq \equiv 3 \mod 4$ and ${\text {pord}}(\rho (\mathfrak {t}))=pq$, then the rank $r \ne \frac{p+q}{2}+1$. Moreover, if $p > 5$, rank $r > \frac{p+q}{2}+1$.

• (6) If we further assume $D^2$ of the modular data or the MTC to be non-integral, then ${\text {pord}}(\tilde{\rho }(\mathfrak {t})) = {\text {ord}}(T) \notin \{2,3,4,6\}$. This implies that ${\text {ord}}(\tilde{\rho }(\mathfrak {t})) \notin \{2,3,4,6\}$.

Some properties of an MD representation depend on the choice of basis. To make use of those properties, we can construct some combinations of $\tilde{\rho }(\mathfrak {s})$s that are invariant under the block-diagonal unitary transformation U.

The eigenvalues of $\tilde{\rho }(\mathfrak {t})$ partition the indices of the basis vectors. To construct the invariant combinations of $\tilde{\rho }(\mathfrak {s})$, for any eigenvalue $\tilde{\theta }$ of $\tilde{\rho }(\mathfrak {t})$, let

$$\begin{aligned} I_{\tilde{\theta }} = \{ i \, \big |\, \tilde{\rho }(\mathfrak {t})_{ii}=\tilde{\theta }\}. \end{aligned}$$

(B.14)

Let $I = I_{\tilde{\theta }}$, $J = J_{\tilde{\theta }'}$, $K = K_{\tilde{\theta }''}$ for some eigenvalues $\tilde{\theta }$, $\tilde{\theta }'$, $\tilde{\theta }''$ of $\tilde{\rho }(\mathfrak {t})$. We see that the following uniform polynomials of $\tilde{\rho }(\mathfrak {s})$ are invariant

$$\begin{aligned} P_I(\rho (\mathfrak {s}))&= {\text {Tr}}\tilde{\rho }(\mathfrak {s})_{II} \equiv \sum _{i\in I} \tilde{\rho }(\mathfrak {s})_{ii}, \nonumber \\ P_{IJ}(\rho (\mathfrak {s}))&= {\text {Tr}}\tilde{\rho }(\mathfrak {s})_{IJ} \tilde{\rho }(\mathfrak {s})_{JI} \equiv \sum _{i\in I,j\in J} \tilde{\rho }(\mathfrak {s})_{i,j} \tilde{\rho }(\mathfrak {s})_{ji}, \nonumber \\ P_{IJK}(\rho (\mathfrak {s}))&= {\text {Tr}}\tilde{\rho }(\mathfrak {s})_{IJ} \tilde{\rho }(\mathfrak {s})_{JK} \tilde{\rho }(\mathfrak {s})_{KI} \equiv \sum _{i\in I,j\in J,k\in K} \tilde{\rho }(\mathfrak {s})_{i,j} \tilde{\rho }(\mathfrak {s})_{j,k} \tilde{\rho }(\mathfrak {s})_{k,i} . \end{aligned}$$

(B.15)

Certainly we can construction many other invariant uniform polynomials in the similar way. Using those invariant uniform polynomials, we have the following results

Proposition B.4

Let $\tilde{\rho }$ be a general ${\text {SL}}_2({\mathbb {Z}})$ representations of a modular data or a MTC. Then following statements hold:

• (1) $\tilde{\rho }(\mathfrak {s})$ satisfies

  $$\begin{aligned} {\text {Tr}}(\tilde{\rho }(\mathfrak {s})^2) \in {\mathbb {Z}}\setminus \{0\} . \end{aligned}$$

  (B.16)

  Let

  $$\begin{aligned} C = \frac{{\text {Tr}}(\tilde{\rho }(\mathfrak {s})^2)}{|{\text {Tr}}(\tilde{\rho }(\mathfrak {s})^2)|} \tilde{\rho }(\mathfrak {s})^2. \end{aligned}$$

  (B.17)

  For all I,

  $$\begin{aligned} P_I(C) \geqslant 0. \end{aligned}$$

  (B.18)

• (2) The conductor of $P_\textrm{odd}(\tilde{\rho }(\mathfrak {s}))$ divides ${\text {ord}}(\tilde{\rho }(\mathfrak {t}))$ for all the invariant uniform polynomials $P_\textrm{odd}$ with odd powers of $\tilde{\rho }(\mathfrak {s})$ (such as $P_I$ and $P_{IJK}$ in (B.15)). The conductor of $P_\textrm{even}(\tilde{\rho }(\mathfrak {s}))$ divides ${\text {pord}}(\tilde{\rho }(\mathfrak {t}))$ for all the invariant uniform polynomials $P_\textrm{even}$ with even powers of $\tilde{\rho }(\mathfrak {s})$ (such as $P_{IJ}$ in (B.15)).

• (3) For any Galois conjugation $\sigma \in {\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\rho (\mathfrak {t}))})$, there is a permutation on the set $\{I\}$, $I \rightarrow {\hat{\sigma }}(I)$, such that

  $$\begin{aligned} \sigma P_{IJ}(\tilde{\rho }(\mathfrak {s}))&= P_{I{\hat{\sigma }}(J)}(\tilde{\rho }(\mathfrak {s})) =P_{{\hat{\sigma }}(I)J}(\tilde{\rho }(\mathfrak {s})) \nonumber \\ \sigma ^2 \big (\tilde{\theta }_I\big )&= \tilde{\theta }_{{\hat{\sigma }} (I)}, \end{aligned}$$

  (B.19)

  for all I, J.

• (4) For any invariant uniform polynomials P (such as those in (B.15))

  $$\begin{aligned} \sigma P\big (\tilde{\rho }(\mathfrak {s})\big ) = P\big (\sigma \tilde{\rho }(\mathfrak {s})\big ) = P \big (\tilde{\rho }(\mathfrak {t})^a \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^b \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^a\big ) \end{aligned}$$

  (B.20)

  where $\sigma \in {\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\tilde{\rho }(\mathfrak {t}))})$, and a, b are given by $\sigma ( \textrm{e}^{\textrm{i}2 \pi /{\text {ord}}(\tilde{\rho }(\mathfrak {t}))} ) = \textrm{e}^{a \textrm{i}2 \pi /{\text {ord}}(\tilde{\rho }(\mathfrak {t}))}$ and $ab \equiv 1$ mod ${\text {ord}}(\tilde{\rho }(\mathfrak {t}))$.

Instead of constructing invariants, there is another way to make use of the properties of an MD representation that depend on the choices of basis. We can choose a more special basis, so that the basis is closer to the basis that leads to the MD representation. For example, we can choose a basis to make $\tilde{\rho }(\mathfrak {s})$ symmetric (i.e. to make $\tilde{\rho }$ a symmetric representation).

Now consider a symmetric ${\text {SL}}_2({\mathbb {Z}})$ matrix representation $\tilde{\rho }$ of a modular data or of a MTC. We find that the restriction of the unitary U in (B.11) on the non-degenerate subspace (see Theorem 3.4) must be diagonal with diagonal elements $U_{ii}\in \{ 1,-1\}$. Therefore, on the non-degenerate subspace, $\tilde{\rho }(\mathfrak {s})$ of a symmetric representation differs from $\rho (\mathfrak {s})$ of an MD representation only by a diagonal unitary transformation U with diagonal elements $\pm 1$, i.e., a signed diagonal matrix. In this case some properties of MD representation apply to the blocks of the symmetric representation within the non-degenerate subspace. This allows us to obtain

Proposition B.5

Let $\tilde{\rho }$ be a symmetric ${\text {SL}}_2({\mathbb {Z}})$ representations equivalent to an MD representation. Let

$$\begin{aligned} I_\textrm{ndeg}&:=\{i \mid \tilde{\rho }(\mathfrak {t})_{i,i} \text { is a non-degenerate eigenvalue}\}, \end{aligned}$$

(B.21)

Then there exists an orthogonal U such that $U \tilde{\rho }U^\top$ is a pMD representation, and the following statements hold:

• (1) The conductor of $(U \tilde{\rho }(\mathfrak {s}) U^\top )_{i,j}$ divides ${\text {ord}}(\tilde{\rho }(\mathfrak {t}))$ for all i, j. This implies that the conductor of $(\tilde{\rho }(\mathfrak {s}))_{i,j}$ divides ${\text {ord}}(\tilde{\rho }(\mathfrak {t}))$ for all $i,j \in I_\textrm{ndeg}$.

• (2) For any Galois conjugation $\sigma$ in ${\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\tilde{\rho }(\mathfrak {t}))})$, there is a permutation $i \rightarrow {\hat{\sigma }}(i)$, such that

  $$\begin{aligned} \sigma \big ((U \tilde{\rho }(\mathfrak {s})U^\top )_{i,j}\big )&= \epsilon _\sigma (i) (U \tilde{\rho }(\mathfrak {s}) U^\top )_{{\hat{\sigma }} (i),j} = (U \tilde{\rho }(\mathfrak {s}) U^\top )_{i,{\hat{\sigma }} (j)} \epsilon _\sigma (j) \nonumber \\ \sigma ^2 \big ( \tilde{\rho }(\mathfrak {t})_{i,i}\big )&= \tilde{\rho }(\mathfrak {t})_{{\hat{\sigma }} (i),{\hat{\sigma }} (i)}, \end{aligned}$$

  (B.22)

  for all i, j, where $\epsilon _\sigma (i)\in \{1,-1\}$. This implies that

  $$\begin{aligned} \sigma \big ( \tilde{\rho }(\mathfrak {s})_{i,j}\big )&= \tilde{\rho }(\mathfrak {s})_{{\hat{\sigma }} (i),j} \ \ \textrm{or} \ \ \sigma \big ( \tilde{\rho }(\mathfrak {s})_{i,j}\big ) = - \tilde{\rho }(\mathfrak {s})_{{\hat{\sigma }} (i),j} \nonumber \\ \sigma \big ( \tilde{\rho }(\mathfrak {s})_{i,j}\big )&= \tilde{\rho }(\mathfrak {s}) _{i,{\hat{\sigma }} (j)} \ \ \textrm{or} \ \ \sigma \big ( \tilde{\rho }(\mathfrak {s})_{i,j}\big ) = - \tilde{\rho }(\mathfrak {s}) _{i,{\hat{\sigma }} (j)} \end{aligned}$$

  (B.23)

  for all $i,j \in I_\textrm{ndeg}$. This also implies that $D_{\tilde{\rho }}(\sigma )$ defined in (3.6) is a signed permutation matrix in the $I_\textrm{ndeg}$ block, i.e. $(D_{\tilde{\rho }}(\sigma ))_{i,j}$ for $i,j \in I_\textrm{ndeg}$ are matrix elements of a signed permutation matrix.

• (3) For all i, j,

  $$\begin{aligned} \sigma \big ((U\tilde{\rho }(\mathfrak {s}) U^\top )_{i,j}\big ) = \big (U \tilde{\rho }(\mathfrak {t})^a \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^b \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^a U^\top \big )_{i,j} \end{aligned}$$

  (B.24)

  where $\sigma \in {\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\tilde{\rho }(\mathfrak {t}))})$, and a, b are given by $\sigma ( \textrm{e}^{\textrm{i}2 \pi /{\text {ord}}(\tilde{\rho }(\mathfrak {t}))} ) = \textrm{e}^{a \textrm{i}2 \pi /{\text {ord}}(\tilde{\rho }(\mathfrak {t}))}$ and $ab \equiv 1$ mod ${\text {ord}}(\tilde{\rho }(\mathfrak {t}))$. This implies that

  $$\begin{aligned} \sigma \big ((\tilde{\rho }(\mathfrak {s}) )_{i,j}\big ) = \big ( \tilde{\rho }(\mathfrak {t})^a \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^b \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^a \big )_{i,j}. \end{aligned}$$

  (B.25)

  for all $i,j \in I_\textrm{ndeg}$.

• (4) Both T and $\tilde{\rho }(\mathfrak {t})$ are diagonal, and without loss of generality, we may assume $\tilde{\rho }(\mathfrak {t})$ is a scalar multiple of T. In this case U in (B.11) is a block diagonal matrix preserving the eigenspaces of $\tilde{\rho }(\mathfrak {t})$. Let $I_\textrm{nonzero} = \{i\}$ be a set of indices such that the $i^\textrm{th}$ row of $U \tilde{\rho }(\mathfrak {s}) U^\top$ contains no zeros for some othorgonal U satisfying $U\tilde{\rho }(\mathfrak {t})U^\top = \tilde{\rho }(\mathfrak {t})$. The index for the unit object of MTC must be in $I_\textrm{nonzero}$. Thus $I_\textrm{nonzero}$ must be nonempty:

  $$\begin{aligned} I_\textrm{nonzero} \ne \varnothing . \end{aligned}$$

  (B.26)

• (5) Let $I_{\tilde{\theta }}$ be a set of indices for an eigenspace $E_{\tilde{\theta }}$ of $\tilde{\rho }(\mathfrak {t})$

  $$\begin{aligned} I_{\tilde{\theta }}&:=\{i \mid \tilde{\rho }(\mathfrak {t})_{i,i} = \tilde{\theta }\}. \end{aligned}$$

  (B.27)

  Then there exists a $I_{\tilde{\theta }}$ such that

  $$\begin{aligned} I_{\tilde{\theta }} \cap I_\textrm{nonzero} \ne \varnothing \ \text { and }\ {\text {Tr}}_{E_{\tilde{\theta }}} C > 0, \end{aligned}$$

  (B.28)

  where C is given in (B.17).

• (6) If we further assume the modular data to be non-integral, then there exists a $I_{\tilde{\theta }}$ that has a non-empty overlap with $I_\textrm{nonzero}$, such that $D_{\tilde{\rho }}(\sigma )_{I_{\tilde{\theta }}} \ne \pm {\text {id}}$ for some $\sigma \in {\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\tilde{\rho }(\mathfrak {t}))}/{\mathbb {Q}})$. Here $D_{\tilde{\rho }}(\sigma )$ is defined in (3.6):

  $$\begin{aligned} D_{\tilde{\rho }}(\sigma )= \tilde{\rho }(\mathfrak {t})^a \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^b \tilde{\rho }(\mathfrak {s}) \tilde{\rho }(\mathfrak {t})^a \tilde{\rho }^{-1}(\mathfrak {s}) \end{aligned}$$

  (B.29)

  where a, b are given by $\sigma ( \textrm{e}^{ 2 \pi \textrm{i}/{\text {ord}}(\tilde{\rho }(\mathfrak {t}))} )=\textrm{e}^{ a 2 \pi \textrm{i}/{\text {ord}}(\tilde{\rho }(\mathfrak {t}))}$ and $ab \equiv 1 \mod {\text {ord}}(\tilde{\rho }(\mathfrak {t}))$. Also $D_{\tilde{\rho }}(\sigma )_{I_{\tilde{\theta }}}$ is the block of $D_{\tilde{\rho }}(\sigma )$ with indices in $I_{\tilde{\theta }}$, i.e. the matrix elements of $D_{\tilde{\rho }}(\sigma )_{I_{\tilde{\theta }}}$ are given by $(D_{\tilde{\rho }}(\sigma ))_{i,j},\ i,j \in I_{\tilde{\theta }}$.

Proposition B.5(6) is a consequence of Theorem 3.13(3). Using GAP System for Computational Discrete Algebra, we obtain a list of symmetric irrep-sum ${\text {SL}}_2({\mathbb {Z}})$ matrix representations that satisfy the conditions in Propositions B.3, B.4, and B.5. The list is given below for rank $r=6$ case (see Appendix section B.2).

Some of those symmetric irrep-sum ${\text {SL}}_2({\mathbb {Z}})$ matrix representations are representations of modular data, while others are not. However, the list includes all the symmetric irrep-sum ${\text {SL}}_2({\mathbb {Z}})$ matrix representations of modular data or MTC’s which are not weakly integral (and some that are weakly integral).

1.2 List of symmetric irrep-sum representations

The following is a list the all rank-6 symmetric irrep-sum representations that satisfy the conditions in Propositions B.3, B.4, and B.5. The list contains all the rank-6 symmetric irrep-sum representations that are unitarily equivalent to rank-6 MD representations, plus some extra ones.

For each symmetric irrep-sum representation, we may generate an orbit by orthogonal transformations

$$\begin{aligned} \rho _\text {isum}(\mathfrak {s}) \rightarrow U \rho _\text {isum}(\mathfrak {s}) U^\top ,\ \ \ \rho _\text {isum}(\mathfrak {t}) \rightarrow U \rho _\text {isum}(\mathfrak {t}) U^\top , \end{aligned}$$

(B.30)

tensoring 1-dimensional ${\text {SL}}_2({\mathbb {Z}})$ representations $\chi _\alpha$, $\alpha =1,\ldots ,12$:

$$\begin{aligned} \rho _\text {isum}(\mathfrak {s}) \rightarrow \chi _\alpha (\mathfrak {s})\rho _\text {isum}(\mathfrak {s}),\ \ \ \rho _\text {isum}(\mathfrak {t}) \rightarrow \chi _\alpha (\mathfrak {t})\rho _\text {isum}(\mathfrak {t}) , \end{aligned}$$

(B.31)

and applying Galois conjugations $\sigma$ in ${\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\rho _\text {isum}(\mathfrak {t}))})$:

$$\begin{aligned} \rho _\text {isum}(\mathfrak {s}) \rightarrow \sigma (\rho _\text {isum}(\mathfrak {s})),\ \ \ \rho _\text {isum}(\mathfrak {t}) \rightarrow \sigma (\rho _\text {isum}(\mathfrak {t})) . \end{aligned}$$

(B.32)

We will call such an orbit a GT orbit. The following list includes only one representative for each GT orbit. The list can also be regarded as a list GT orbits.

In the list, a representation $\rho _\text {isum}$ is expressed as the direct sum of irreducible representations $\rho _\text {isum} = \rho _1\oplus \rho _2 \oplus \cdots$, where $\rho _a(\mathfrak {t})$ is presented as $({\tilde{s}}_{1},{\tilde{s}}_{2},\cdots )$ with ${\tilde{s}}_{i} = \arg (\rho _a(\mathfrak {t})_{ii})$, and $\rho _a(\mathfrak {s})$ is presented as $(\rho _{11}, \rho _{12}, \rho _{13}, \rho _{14}, \cdots ;\ \ \rho _{22}, \rho _{23}, \rho _{24}, \cdots )$. The direct sum is also given via an index form, for example, irreps = $2_{2}^{1,0} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{5}^{1,0}\oplus 2_{5}^{1,0}$. It means that the representation $\rho _\text {isum}$ is a direct sum of two irreducible representations $2_{2}^{1,0} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{5}^{1,0}$ and $2_{5}^{1,0}$. Here $2_{2}^{1,0}$, $2_{5}^{1,0}$ are indices of ${\text {SL}}_2({\mathbb {Z}})$ irreducible representations with prime-power levels. Those prime-power-level ${\text {SL}}_2({\mathbb {Z}})$ irreducible representation are listed in “Appendix A”, where the meaning of the indices is explained further. $2_{2}^{1,0} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{5}^{1,0}$ is the irreducible representation obtained by the tensor product of $2_{2}^{1,0}$ and $2_{5}^{1,0}$.

The dimensions of the representations $\rho _\text {isum}$ are given by dims = $(r_1, r_2, \cdots )$, where $r_a$ is the dimension of the irreducible representation $\rho _a$, satisfying $r_1\geqslant r_2 \geqslant \cdots$. The levels of the representations $\rho _a$ are given by levels = $(l_1, l_2, \cdots )$, where $l_a =\textrm{ord}(\rho _a(\mathfrak {t}))$. We will use (dims;levels) = $(r_1, r_2, \cdots ; l_1,l_2,\cdots )$ to label those representations. Now we can explain how the representative of a GT orbit is chosen. The representative for a GT orbit is chosen to be the one with minimal $[ [r_1,r_2,\cdots ], {\text {ord}}(\rho _\text {isum}(\mathfrak {t})), [l_1,l_2,\cdots ] ]$. Here the order of two lists is determined by first compare the first elements of the two lists. If the first elements are equal, we then compare the second elements, etc. The order of cyclotomic numbers are given by GAP.

To describe the entries of $\rho _a(\mathfrak {s})$, we also introduced the following notations:

$$\begin{aligned} \zeta ^m_n&=\textrm{e}^{2\pi \textrm{i} m/n} ,\ \ \ c^m_n = \zeta ^m_n+\zeta ^{-m}_n ,\ \ \ s^m_n = \zeta ^m_n-\zeta ^{-m}_n, \nonumber \\ \xi ^{m,k}_n&= (\zeta ^m_{2n}-\zeta ^{-m}_{2n})/(\zeta _{2n}^k-\zeta _{2n}^{-k}),\ \ \ \ \xi ^{m}_n = \xi ^{m,1}_n . \end{aligned}$$

(B.33)

We find that, for rank 6, there are only 25 GT orbits. The GT orbits can be divided into two classes, resolved and unresolved, whose definition will to given in the next section. Below each GT orbit, we indicate whether it is resolved or unresolved. Among 25 GT orbits, 17 are resolved and 8 are unresolved.

For the 17 resolved GT orbits, it is easy to compute all the corresponding pairs of (S, T) matrices that satisfied the conditions in Proposition B.1, which will be done in next section. Below each resolved GT orbit, we indicate the number valid (S, T) pairs obtain with such a computation. Those valid (S, T) pairs will be listed in “Appendix C.2”. The 8 unresolved GT orbits are difficult to handle by computer, which are discussed in the main text. (The main text also discussed most of the resolved cases.)

• 1. (dims;levels) =(3, 2, 1; 5, 5, 1), irreps = $3_{5}^{1}\oplus 2_{5}^{1}\oplus 1_{1}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 5$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{5}, \frac{4}{5} ) \oplus ( \frac{1}{5}, \frac{4}{5} ) \oplus ( 0 )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\sqrt{\frac{1}{5}}$, $-\sqrt{\frac{2}{5}}$, $-\sqrt{\frac{2}{5}}$; $-\frac{5+\sqrt{5}}{10}$, $\frac{5-\sqrt{5}}{10}$; $-\frac{5+\sqrt{5}}{10}$) $\oplus$ $\textrm{i}$($-\frac{1}{\sqrt{5}}c^{3}_{20}$, $\frac{1}{\sqrt{5}}c^{1}_{20}$; $\frac{1}{\sqrt{5}}c^{3}_{20}$) $\oplus$ (1) Resolved. Number of valid (S, T) pairs = 0.

• 2. (dims;levels) =(3, 2, 1; 8, 8, 1), irreps = $3_{8}^{1,0}\oplus 2_{8}^{1,9}\oplus 1_{1}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 8$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{8}, \frac{5}{8} ) \oplus ( \frac{1}{8}, \frac{7}{8} ) \oplus ( 0 )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$(0, $\sqrt{\frac{1}{2}}$, $\sqrt{\frac{1}{2}}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$) $\oplus$ $\textrm{i}$($-\sqrt{\frac{1}{2}}$, $\sqrt{\frac{1}{2}}$; $\sqrt{\frac{1}{2}}$) $\oplus$ (1) Resolved. Number of valid (S, T) pairs = 0.

• 3. (dims;levels) =(3, 2, 1; 5, 2, 1), irreps = $3_{5}^{1}\oplus 2_{2}^{1,0}\oplus 1_{1}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 10$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{5}, \frac{4}{5} ) \oplus ( 0, \frac{1}{2} ) \oplus ( 0 )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\sqrt{\frac{1}{5}}$, $-\sqrt{\frac{2}{5}}$, $-\sqrt{\frac{2}{5}}$; $-\frac{5+\sqrt{5}}{10}$, $\frac{5-\sqrt{5}}{10}$; $-\frac{5+\sqrt{5}}{10}$) $\oplus$ ($-\frac{1}{2}$, $-\sqrt{\frac{3}{4}}$; $\frac{1}{2}$) $\oplus$ (1) Unresolved.

• 4. (dims;levels) =(3, 2, 1; 5, 2, 2), irreps = $3_{5}^{1}\oplus 2_{2}^{1,0}\oplus 1_{2}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 10$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{5}, \frac{4}{5} ) \oplus ( 0, \frac{1}{2} ) \oplus ( \frac{1}{2} )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\sqrt{\frac{1}{5}}$, $-\sqrt{\frac{2}{5}}$, $-\sqrt{\frac{2}{5}}$; $-\frac{5+\sqrt{5}}{10}$, $\frac{5-\sqrt{5}}{10}$; $-\frac{5+\sqrt{5}}{10}$) $\oplus$ ($-\frac{1}{2}$, $-\sqrt{\frac{3}{4}}$; $\frac{1}{2}$) $\oplus$ ($-1$) Unresolved.

• 5. (dims;levels) =(3, 2, 1; 4, 3, 2), irreps = $3_{4}^{1,3}\oplus 2_{3}^{1,0}\oplus 1_{2}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 12$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{2}, \frac{1}{4} ) \oplus ( 0, \frac{1}{3} ) \oplus ( \frac{1}{2} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($-\frac{1}{2}$, $\frac{1}{2}$, $\sqrt{\frac{1}{2}}$; $-\frac{1}{2}$, $\sqrt{\frac{1}{2}}$; 0) $\oplus$ $\textrm{i}$($-\sqrt{\frac{1}{3}}$, $\sqrt{\frac{2}{3}}$; $\sqrt{\frac{1}{3}}$) $\oplus$ ($-1$) Resolved. Number of valid (S, T) pairs = 0.

• 6. (dims;levels) =(3, 2, 1; 4, 3, 4), irreps = $3_{4}^{1,3}\oplus 2_{3}^{1,0}\oplus 1_{4}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 12$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{2}, \frac{1}{4} ) \oplus ( 0, \frac{1}{3} ) \oplus ( \frac{1}{4} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($-\frac{1}{2}$, $\frac{1}{2}$, $\sqrt{\frac{1}{2}}$; $-\frac{1}{2}$, $\sqrt{\frac{1}{2}}$; 0) $\oplus$ $\textrm{i}$($-\sqrt{\frac{1}{3}}$, $\sqrt{\frac{2}{3}}$; $\sqrt{\frac{1}{3}}$) $\oplus$ $\textrm{i}$(1) Unresolved.

• 7. (dims;levels) =(3, 2, 1; 8, 3, 1), irreps = $3_{8}^{1,0}\oplus 2_{3}^{1,0}\oplus 1_{1}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 24$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{8}, \frac{5}{8} ) \oplus ( 0, \frac{1}{3} ) \oplus ( 0 )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$(0, $\sqrt{\frac{1}{2}}$, $\sqrt{\frac{1}{2}}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$) $\oplus$ $\textrm{i}$($-\sqrt{\frac{1}{3}}$, $\sqrt{\frac{2}{3}}$; $\sqrt{\frac{1}{3}}$) $\oplus$ (1) Resolved. Number of valid (S, T) pairs = 0.

• 8. (dims;levels) =(3, 2, 1; 8, 3, 3), irreps = $3_{8}^{1,0}\oplus 2_{3}^{1,0}\oplus 1_{3}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 24$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{8}, \frac{5}{8} ) \oplus ( 0, \frac{1}{3} ) \oplus ( \frac{1}{3} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$(0, $\sqrt{\frac{1}{2}}$, $\sqrt{\frac{1}{2}}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$) $\oplus$ $\textrm{i}$($-\sqrt{\frac{1}{3}}$, $\sqrt{\frac{2}{3}}$; $\sqrt{\frac{1}{3}}$) $\oplus$ (1) Resolved. Number of valid (S, T) pairs = 0.

• 9. (dims;levels) =(3, 3; 5, 3), irreps = $3_{5}^{1}\oplus 3_{3}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 15$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{5}, \frac{4}{5} ) \oplus ( 0, \frac{1}{3}, \frac{2}{3} )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\sqrt{\frac{1}{5}}$, $-\sqrt{\frac{2}{5}}$, $-\sqrt{\frac{2}{5}}$; $-\frac{5+\sqrt{5}}{10}$, $\frac{5-\sqrt{5}}{10}$; $-\frac{5+\sqrt{5}}{10}$) $\oplus$ ($-\frac{1}{3}$, $\frac{2}{3}$, $\frac{2}{3}$; $-\frac{1}{3}$, $\frac{2}{3}$; $-\frac{1}{3}$) Resolved. Number of valid (S, T) pairs = 0.

• 10. (dims;levels) =(3, 3; 16, 16), irreps = $3_{16}^{1,0}\oplus 3_{16}^{1,6}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 16$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{8}, \frac{1}{16}, \frac{9}{16} ) \oplus ( \frac{5}{8}, \frac{1}{16}, \frac{9}{16} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$(0, $\sqrt{\frac{1}{2}}$, $\sqrt{\frac{1}{2}}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$) $\oplus$ $\textrm{i}$(0, $\sqrt{\frac{1}{2}}$, $\sqrt{\frac{1}{2}}$; $\frac{1}{2}$, $-\frac{1}{2}$; $\frac{1}{2}$) Unresolved.

• 11. (dims;levels) =(3, 3; 5, 4), irreps = $3_{5}^{1}\oplus 3_{4}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 20$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{5}, \frac{4}{5} ) \oplus ( 0, \frac{1}{4}, \frac{3}{4} )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\sqrt{\frac{1}{5}}$, $-\sqrt{\frac{2}{5}}$, $-\sqrt{\frac{2}{5}}$; $-\frac{5+\sqrt{5}}{10}$, $\frac{5-\sqrt{5}}{10}$; $-\frac{5+\sqrt{5}}{10}$) $\oplus$ (0, $\sqrt{\frac{1}{2}}$, $\sqrt{\frac{1}{2}}$; $-\frac{1}{2}$, $\frac{1}{2}$; $-\frac{1}{2}$) Resolved. Number of valid (S, T) pairs = 2.

• 12. (dims;levels) =(4, 1, 1; 9, 1, 1), irreps = $4_{9,2}^{1,0}\oplus 1_{1}^{1}\oplus 1_{1}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 9$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{9}, \frac{4}{9}, \frac{7}{9} ) \oplus ( 0 ) \oplus ( 0 )$, $\rho _\text {isum}(\mathfrak {s})$ = (0, $-\sqrt{\frac{1}{3}}$, $-\sqrt{\frac{1}{3}}$, $-\sqrt{\frac{1}{3}}$; $\frac{1}{3}c^{2}_{9}$, $\frac{1}{3} c_9^4$, $\frac{1}{3}c^{1}_{9}$; $\frac{1}{3}c^{1}_{9}$, $\frac{1}{3}c^{2}_{9}$; $\frac{1}{3} c_9^4$) $\oplus$ (1) $\oplus$ (1) Unresolved.

• 13. (dims;levels) =(4, 2; 5, 5), irreps = $4_{5,1}^{1}\oplus 2_{5}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 5$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ) \oplus ( \frac{1}{5}, \frac{4}{5} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($\frac{1}{5}c^{1}_{20} +\frac{1}{5}c^{3}_{20}$, $\frac{2}{5}c^{2}_{15} +\frac{1}{5}c^{3}_{15}$, $-\frac{1}{5}+\frac{2}{5}c^{1}_{15} -\frac{1}{5}c^{3}_{15}$, $\frac{1}{5}c^{1}_{20} -\frac{1}{5}c^{3}_{20}$; $-\frac{1}{5}c^{1}_{20} +\frac{1}{5}c^{3}_{20}$, $-\frac{1}{5}c^{1}_{20} -\frac{1}{5}c^{3}_{20}$, $\frac{1}{5}-\frac{2}{5}c^{1}_{15} +\frac{1}{5}c^{3}_{15}$; $\frac{1}{5}c^{1}_{20} -\frac{1}{5}c^{3}_{20}$, $\frac{2}{5}c^{2}_{15} +\frac{1}{5}c^{3}_{15}$; $-\frac{1}{5}c^{1}_{20} -\frac{1}{5}c^{3}_{20}$) $\oplus$ $\textrm{i}$($-\frac{1}{\sqrt{5}}c^{3}_{20}$, $\frac{1}{\sqrt{5}}c^{1}_{20}$; $\frac{1}{\sqrt{5}}c^{3}_{20}$) Unresolved.

• 14. (dims;levels) =(4, 2; 5, 5; a), irreps = $4_{5,2}^{1}\oplus 2_{5}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 5$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ) \oplus ( \frac{1}{5}, \frac{4}{5} )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\sqrt{\frac{1}{5}}$, $\frac{-5+\sqrt{5}}{10}$, $-\frac{5+\sqrt{5}}{10}$, $\sqrt{\frac{1}{5}}$; $-\sqrt{\frac{1}{5}}$, $\sqrt{\frac{1}{5}}$, $\frac{5+\sqrt{5}}{10}$; $-\sqrt{\frac{1}{5}}$, $\frac{5-\sqrt{5}}{10}$; $\sqrt{\frac{1}{5}}$) $\oplus$ $\textrm{i}$($-\frac{1}{\sqrt{5}}c^{3}_{20}$, $\frac{1}{\sqrt{5}}c^{1}_{20}$; $\frac{1}{\sqrt{5}}c^{3}_{20}$) Resolved. Number of valid (S, T) pairs = 0.

• 15. (dims;levels) =(4, 2; 10, 5), irreps = $2_{5}^{1} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{2}^{1,0}\oplus 2_{5}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 10$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{5}, \frac{4}{5}, \frac{3}{10}, \frac{7}{10} ) \oplus ( \frac{1}{5}, \frac{4}{5} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($\frac{1}{2\sqrt{5}}c^{3}_{20}$, $\frac{1}{2\sqrt{5}}c^{1}_{20}$, $\frac{3}{2\sqrt{15}}c^{1}_{20}$, $\frac{3}{2\sqrt{15}}c^{3}_{20}$; $-\frac{1}{2\sqrt{5}}c^{3}_{20}$, $-\frac{3}{2\sqrt{15}}c^{3}_{20}$, $\frac{3}{2\sqrt{15}}c^{1}_{20}$; $\frac{1}{2\sqrt{5}}c^{3}_{20}$, $-\frac{1}{2\sqrt{5}}c^{1}_{20}$; $-\frac{1}{2\sqrt{5}}c^{3}_{20}$) $\oplus$ $\textrm{i}$($-\frac{1}{\sqrt{5}}c^{3}_{20}$, $\frac{1}{\sqrt{5}}c^{1}_{20}$; $\frac{1}{\sqrt{5}}c^{3}_{20}$) Unresolved.

• 16. (dims;levels) =(4, 2; 15, 5), irreps = $2_{5}^{1} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{3}^{1,0}\oplus 2_{5}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 15$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{5}, \frac{4}{5}, \frac{2}{15}, \frac{8}{15} ) \oplus ( \frac{1}{5}, \frac{4}{5} )$, $\rho _\text {isum}(\mathfrak {s})$ = ($-\frac{1}{\sqrt{15}}c^{3}_{20}$, $\frac{1}{\sqrt{15}}c^{1}_{20}$, $\frac{2}{\sqrt{30}}c^{1}_{20}$, $-\frac{2}{\sqrt{30}}c^{3}_{20}$; $\frac{1}{\sqrt{15}}c^{3}_{20}$, $\frac{2}{\sqrt{30}}c^{3}_{20}$, $\frac{2}{\sqrt{30}}c^{1}_{20}$; $-\frac{1}{\sqrt{15}}c^{3}_{20}$, $-\frac{1}{\sqrt{15}}c^{1}_{20}$; $\frac{1}{\sqrt{15}}c^{3}_{20}$) $\oplus$ $\textrm{i}$($-\frac{1}{\sqrt{5}}c^{3}_{20}$, $\frac{1}{\sqrt{5}}c^{1}_{20}$; $\frac{1}{\sqrt{5}}c^{3}_{20}$) Resolved. Number of valid (S, T) pairs = 1.

• 17. (dims;levels) =(4, 2; 7, 3), irreps = $4_{7}^{1}\oplus 2_{3}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 21$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{7}, \frac{2}{7}, \frac{4}{7} ) \oplus ( 0, \frac{1}{3} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($-\sqrt{\frac{1}{7}}$, $\sqrt{\frac{2}{7}}$, $\sqrt{\frac{2}{7}}$, $\sqrt{\frac{2}{7}}$; $-\frac{1}{\sqrt{7}}c^{2}_{7}$, $-\frac{1}{\sqrt{7}}c^{1}_{7}$, $\frac{1}{\sqrt{7}\textrm{i}}s^{5}_{28}$; $\frac{1}{\sqrt{7}\textrm{i}}s^{5}_{28}$, $-\frac{1}{\sqrt{7}}c^{2}_{7}$; $-\frac{1}{\sqrt{7}}c^{1}_{7}$) $\oplus$ $\textrm{i}$($-\sqrt{\frac{1}{3}}$, $\sqrt{\frac{2}{3}}$; $\sqrt{\frac{1}{3}}$) Resolved. Number of valid (S, T) pairs = 1.

• 18. (dims;levels) =(5, 1; 5, 1), irreps = $5_{5}^{1}\oplus 1_{1}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 5$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ) \oplus ( 0 )$, $\rho _\text {isum}(\mathfrak {s})$ = ($-\frac{1}{5}$, $\sqrt{\frac{6}{25}}$, $\sqrt{\frac{6}{25}}$, $\sqrt{\frac{6}{25}}$, $\sqrt{\frac{6}{25}}$; $\frac{3-\sqrt{5}}{10}$, $-\frac{1+\sqrt{5}}{5}$, $\frac{-1+\sqrt{5}}{5}$, $\frac{3+\sqrt{5}}{10}$; $\frac{3+\sqrt{5}}{10}$, $\frac{3-\sqrt{5}}{10}$, $\frac{-1+\sqrt{5}}{5}$; $\frac{3+\sqrt{5}}{10}$, $-\frac{1+\sqrt{5}}{5}$; $\frac{3-\sqrt{5}}{10}$) $\oplus$ (1) Unresolved.

• 19. (dims;levels) =(6; 9), irreps = $6_{9,3}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 9$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{9}, \frac{2}{9}, \frac{4}{9}, \frac{5}{9}, \frac{7}{9}, \frac{8}{9} )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\frac{1}{3}$, $\frac{1}{3}c^{2}_{9}$, $\frac{1}{3}$, $-\frac{1}{3}c^{1}_{9}$, $\frac{1}{3}$, $\frac{1}{3} c_9^4$; $\frac{1}{3}$, $\frac{1}{3} c_9^4$, $-\frac{1}{3}$, $\frac{1}{3}c^{1}_{9}$, $\frac{1}{3}$; $\frac{1}{3}$, $-\frac{1}{3}c^{2}_{9}$, $\frac{1}{3}$, $\frac{1}{3}c^{1}_{9}$; $\frac{1}{3}$, $-\frac{1}{3} c_9^4$, $-\frac{1}{3}$; $\frac{1}{3}$, $\frac{1}{3}c^{2}_{9}$; $\frac{1}{3}$) Resolved. Number of valid (S, T) pairs = 1.

• 20. (dims;levels) =(6; 13), irreps = $6_{13}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 13$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{13}, \frac{3}{13}, \frac{4}{13}, \frac{9}{13}, \frac{10}{13}, \frac{12}{13} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($-\frac{1}{\sqrt{13}}c^{5}_{52}$, $\frac{1}{\sqrt{13}}c^{7}_{52}$, $\frac{1}{\sqrt{13}}c^{3}_{52}$, $\frac{1}{\sqrt{13}}c^{11}_{52}$, $\frac{1}{\sqrt{13}}c^{9}_{52}$, $-\frac{1}{\sqrt{13}}c^{1}_{52}$; $-\frac{1}{\sqrt{13}}c^{11}_{52}$, $\frac{1}{\sqrt{13}}c^{1}_{52}$, $-\frac{1}{\sqrt{13}}c^{5}_{52}$, $\frac{1}{\sqrt{13}}c^{3}_{52}$, $\frac{1}{\sqrt{13}}c^{9}_{52}$; $\frac{1}{\sqrt{13}}c^{7}_{52}$, $\frac{1}{\sqrt{13}}c^{9}_{52}$, $-\frac{1}{\sqrt{13}}c^{5}_{52}$, $\frac{1}{\sqrt{13}}c^{11}_{52}$; $-\frac{1}{\sqrt{13}}c^{7}_{52}$, $-\frac{1}{\sqrt{13}}c^{1}_{52}$, $-\frac{1}{\sqrt{13}}c^{3}_{52}$; $\frac{1}{\sqrt{13}}c^{11}_{52}$, $-\frac{1}{\sqrt{13}}c^{7}_{52}$; $\frac{1}{\sqrt{13}}c^{5}_{52}$) Resolved. Number of valid (S, T) pairs = 1.

• 21. (dims;levels) =(6; 15), irreps = $3_{3}^{1,0} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{5}^{1}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 15$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{5}, \frac{4}{5}, \frac{2}{15}, \frac{7}{15}, \frac{8}{15}, \frac{13}{15} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($\frac{1}{3\sqrt{5}}c^{3}_{20}$, $\frac{1}{3\sqrt{5}}c^{1}_{20}$, $\frac{2}{3\sqrt{5}}c^{1}_{20}$, $\frac{2}{3\sqrt{5}}c^{1}_{20}$, $\frac{2}{3\sqrt{5}}c^{3}_{20}$, $\frac{2}{3\sqrt{5}}c^{3}_{20}$; $-\frac{1}{3\sqrt{5}}c^{3}_{20}$, $-\frac{2}{3\sqrt{5}}c^{3}_{20}$, $-\frac{2}{3\sqrt{5}}c^{3}_{20}$, $\frac{2}{3\sqrt{5}}c^{1}_{20}$, $\frac{2}{3\sqrt{5}}c^{1}_{20}$; $-\frac{1}{3\sqrt{5}}c^{3}_{20}$, $\frac{2}{3\sqrt{5}}c^{3}_{20}$, $\frac{1}{3\sqrt{5}}c^{1}_{20}$, $-\frac{2}{3\sqrt{5}}c^{1}_{20}$; $-\frac{1}{3\sqrt{5}}c^{3}_{20}$, $-\frac{2}{3\sqrt{5}}c^{1}_{20}$, $\frac{1}{3\sqrt{5}}c^{1}_{20}$; $\frac{1}{3\sqrt{5}}c^{3}_{20}$, $-\frac{2}{3\sqrt{5}}c^{3}_{20}$; $\frac{1}{3\sqrt{5}}c^{3}_{20}$) Resolved. Number of valid (S, T) pairs = 0.

• 22. (dims;levels) =(6; 16), irreps = $6_{16,1}^{1,0}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 16$, $\rho _\text {isum}(\mathfrak {t})$ = $( 0, \frac{1}{4}, \frac{1}{16}, \frac{5}{16}, \frac{9}{16}, \frac{13}{16} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$(0, 0, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$, $\frac{1}{2}$; 0, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{1}{2}$, $-\frac{1}{2}$; $-\sqrt{\frac{1}{8}}$, $-\sqrt{\frac{1}{8}}$, $\sqrt{\frac{1}{8}}$, $\sqrt{\frac{1}{8}}$; $\sqrt{\frac{1}{8}}$, $\sqrt{\frac{1}{8}}$, $-\sqrt{\frac{1}{8}}$; $-\sqrt{\frac{1}{8}}$, $-\sqrt{\frac{1}{8}}$; $\sqrt{\frac{1}{8}}$) Resolved. Number of valid (S, T) pairs = 4.

• 23. (dims;levels) =(6; 35), irreps = $3_{7}^{3} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{5}^{2}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 35$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{35}, \frac{4}{35}, \frac{9}{35}, \frac{11}{35}, \frac{16}{35}, \frac{29}{35} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$($-\frac{4}{35}c^{1}_{140} -\frac{3}{35}c^{3}_{140} -\frac{1}{7}c^{5}_{140} +\frac{1}{35}c^{7}_{140} +\frac{1}{35}c^{9}_{140} +\frac{4}{35}c^{13}_{140} +\frac{2}{35}c^{15}_{140} -\frac{3}{35}c^{17}_{140} +\frac{9}{35}c^{19}_{140} -\frac{4}{35}c^{21}_{140} -\frac{2}{7}c^{23}_{140}$, $-\frac{1}{\sqrt{35}}c^{4}_{35} +\frac{1}{\sqrt{35}}c^{11}_{35}$, $\frac{1}{\sqrt{35}}c^{1}_{35} -\frac{1}{\sqrt{35}}c^{6}_{35}$, $\frac{2}{\sqrt{35}}c^{3}_{35} +\frac{1}{\sqrt{35}}c^{4}_{35} +\frac{1}{\sqrt{35}}c^{10}_{35} +\frac{1}{\sqrt{35}}c^{11}_{35}$, $-\frac{1}{\sqrt{35}\textrm{i}}s^{3}_{140} -\frac{1}{\sqrt{35}\textrm{i}}s^{17}_{140}$, $\frac{2}{35}c^{1}_{140} -\frac{1}{35}c^{3}_{140} -\frac{1}{7}c^{5}_{140} -\frac{3}{35}c^{7}_{140} +\frac{1}{5}c^{9}_{140} -\frac{2}{35}c^{13}_{140} -\frac{1}{35}c^{15}_{140} -\frac{1}{35}c^{17}_{140} +\frac{3}{35}c^{19}_{140} +\frac{2}{35}c^{21}_{140} -\frac{2}{7}c^{23}_{140}$; $-\frac{1}{\sqrt{35}\textrm{i}}s^{3}_{140} -\frac{1}{\sqrt{35}\textrm{i}}s^{17}_{140}$, $\frac{4}{35}c^{1}_{140} +\frac{3}{35}c^{3}_{140} +\frac{1}{7}c^{5}_{140} -\frac{1}{35}c^{7}_{140} -\frac{1}{35}c^{9}_{140} -\frac{4}{35}c^{13}_{140} -\frac{2}{35}c^{15}_{140} +\frac{3}{35}c^{17}_{140} -\frac{9}{35}c^{19}_{140} +\frac{4}{35}c^{21}_{140} +\frac{2}{7}c^{23}_{140}$, $-\frac{1}{\sqrt{35}}c^{1}_{35} +\frac{1}{\sqrt{35}}c^{6}_{35}$, $-\frac{2}{35}c^{1}_{140} +\frac{1}{35}c^{3}_{140} +\frac{1}{7}c^{5}_{140} +\frac{3}{35}c^{7}_{140} -\frac{1}{5}c^{9}_{140} +\frac{2}{35}c^{13}_{140} +\frac{1}{35}c^{15}_{140} +\frac{1}{35}c^{17}_{140} -\frac{3}{35}c^{19}_{140} -\frac{2}{35}c^{21}_{140} +\frac{2}{7}c^{23}_{140}$, $\frac{2}{\sqrt{35}}c^{3}_{35} +\frac{1}{\sqrt{35}}c^{4}_{35} +\frac{1}{\sqrt{35}}c^{10}_{35} +\frac{1}{\sqrt{35}}c^{11}_{35}$; $\frac{2}{\sqrt{35}}c^{3}_{35} +\frac{1}{\sqrt{35}}c^{4}_{35} +\frac{1}{\sqrt{35}}c^{10}_{35} +\frac{1}{\sqrt{35}}c^{11}_{35}$, $-\frac{2}{35}c^{1}_{140} +\frac{1}{35}c^{3}_{140} +\frac{1}{7}c^{5}_{140} +\frac{3}{35}c^{7}_{140} -\frac{1}{5}c^{9}_{140} +\frac{2}{35}c^{13}_{140} +\frac{1}{35}c^{15}_{140} +\frac{1}{35}c^{17}_{140} -\frac{3}{35}c^{19}_{140} -\frac{2}{35}c^{21}_{140} +\frac{2}{7}c^{23}_{140}$, $\frac{1}{\sqrt{35}}c^{4}_{35} -\frac{1}{\sqrt{35}}c^{11}_{35}$, $-\frac{1}{\sqrt{35}\textrm{i}}s^{3}_{140} -\frac{1}{\sqrt{35}\textrm{i}}s^{17}_{140}$; $\frac{1}{\sqrt{35}\textrm{i}}s^{3}_{140} +\frac{1}{\sqrt{35}\textrm{i}}s^{17}_{140}$, $-\frac{4}{35}c^{1}_{140} -\frac{3}{35}c^{3}_{140} -\frac{1}{7}c^{5}_{140} +\frac{1}{35}c^{7}_{140} +\frac{1}{35}c^{9}_{140} +\frac{4}{35}c^{13}_{140} +\frac{2}{35}c^{15}_{140} -\frac{3}{35}c^{17}_{140} +\frac{9}{35}c^{19}_{140} -\frac{4}{35}c^{21}_{140} -\frac{2}{7}c^{23}_{140}$, $\frac{1}{\sqrt{35}}c^{4}_{35} -\frac{1}{\sqrt{35}}c^{11}_{35}$; $-\frac{2}{\sqrt{35}}c^{3}_{35} -\frac{1}{\sqrt{35}}c^{4}_{35} -\frac{1}{\sqrt{35}}c^{10}_{35} -\frac{1}{\sqrt{35}}c^{11}_{35}$, $-\frac{1}{\sqrt{35}}c^{1}_{35} +\frac{1}{\sqrt{35}}c^{6}_{35}$; $\frac{4}{35}c^{1}_{140} +\frac{3}{35}c^{3}_{140} +\frac{1}{7}c^{5}_{140} -\frac{1}{35}c^{7}_{140} -\frac{1}{35}c^{9}_{140} -\frac{4}{35}c^{13}_{140} -\frac{2}{35}c^{15}_{140} +\frac{3}{35}c^{17}_{140} -\frac{9}{35}c^{19}_{140} +\frac{4}{35}c^{21}_{140} +\frac{2}{7}c^{23}_{140}$) Resolved. Number of valid (S, T) pairs = 1.

• 24. (dims;levels) =(6; 56), irreps = $3_{7}^{1} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{8}^{1,6}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 28$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{56}, \frac{9}{56}, \frac{11}{56}, \frac{25}{56}, \frac{43}{56}, \frac{51}{56} )$, $\rho _\text {isum}(\mathfrak {s})$ = ($\frac{1}{\sqrt{14}}c^{1}_{28}$, $\frac{1}{\sqrt{14}}c^{3}_{28}$, $-\frac{1}{\sqrt{14}}c^{5}_{28}$, $-\frac{1}{\sqrt{14}}c^{5}_{28}$, $\frac{1}{\sqrt{14}}c^{1}_{28}$, $\frac{1}{\sqrt{14}}c^{3}_{28}$; $-\frac{1}{\sqrt{14}}c^{5}_{28}$, $\frac{1}{\sqrt{14}}c^{1}_{28}$, $\frac{1}{\sqrt{14}}c^{1}_{28}$, $\frac{1}{\sqrt{14}}c^{3}_{28}$, $-\frac{1}{\sqrt{14}}c^{5}_{28}$; $-\frac{1}{\sqrt{14}}c^{3}_{28}$, $\frac{1}{\sqrt{14}}c^{3}_{28}$, $\frac{1}{\sqrt{14}}c^{5}_{28}$, $-\frac{1}{\sqrt{14}}c^{1}_{28}$; $\frac{1}{\sqrt{14}}c^{3}_{28}$, $-\frac{1}{\sqrt{14}}c^{5}_{28}$, $\frac{1}{\sqrt{14}}c^{1}_{28}$; $-\frac{1}{\sqrt{14}}c^{1}_{28}$, $-\frac{1}{\sqrt{14}}c^{3}_{28}$; $\frac{1}{\sqrt{14}}c^{5}_{28}$) Resolved. Number of valid (S, T) pairs = 2.

• 25. (dims;levels) =(6; 80), irreps = $3_{16}^{3,3} \hspace{-1.5pt}\otimes \hspace{-1.5pt}2_{5}^{2}$, pord$(\rho _\text {isum}(\mathfrak {t})) = 80$, $\rho _\text {isum}(\mathfrak {t})$ = $( \frac{1}{40}, \frac{9}{40}, \frac{3}{80}, \frac{27}{80}, \frac{43}{80}, \frac{67}{80} )$, $\rho _\text {isum}(\mathfrak {s})$ = $\textrm{i}$(0, 0, $\frac{1}{\sqrt{10}}c^{3}_{20}$, $\frac{1}{\sqrt{10}}c^{1}_{20}$, $\frac{1}{\sqrt{10}}c^{3}_{20}$, $\frac{1}{\sqrt{10}}c^{1}_{20}$; 0, $\frac{1}{\sqrt{10}}c^{1}_{20}$, $-\frac{1}{\sqrt{10}}c^{3}_{20}$, $\frac{1}{\sqrt{10}}c^{1}_{20}$, $-\frac{1}{\sqrt{10}}c^{3}_{20}$; $-\frac{1}{2\sqrt{5}}c^{1}_{20}$, $-\frac{1}{2\sqrt{5}}c^{3}_{20}$, $\frac{1}{2\sqrt{5}}c^{1}_{20}$, $\frac{1}{2\sqrt{5}}c^{3}_{20}$; $\frac{1}{2\sqrt{5}}c^{1}_{20}$, $\frac{1}{2\sqrt{5}}c^{3}_{20}$, $-\frac{1}{2\sqrt{5}}c^{1}_{20}$; $-\frac{1}{2\sqrt{5}}c^{1}_{20}$, $-\frac{1}{2\sqrt{5}}c^{3}_{20}$; $\frac{1}{2\sqrt{5}}c^{1}_{20}$) Resolved. Number of valid (S, T) pairs = 2.

A List of Candidate Modular Data from Resolved ${\text {SL}}_2({\mathbb {Z}})$ Representations

1.1 The notion of resolved ${\text {SL}}_2({\mathbb {Z}})$ matrix representations

In the above, we have chosen a special basis in the eigenspaces of an ${\text {SL}}_2({\mathbb {Z}})$ matrix representation $\tilde{\rho }$ to make $\tilde{\rho }(\mathfrak {s})$ symmetric. But such a special basis is still not special enough to make $\tilde{\rho }$ to be an MD representation $\rho$.

We can choose a more special basis to make $\tilde{\rho }(\mathfrak {s}^2)$ a signed permutation matrix, and $\tilde{\rho }(\mathfrak {s})$ symmetric. We know that, for an MD representation $\rho$, $\rho (\mathfrak {s}^2)$ is a signed permutation matrix. So the new special basis makes $\tilde{\rho }$ closer to the MD representation $\rho$.

We can choose an even more special basis in the eigenspaces of $\tilde{\rho }(\mathfrak {t})$ to make $\tilde{\rho }$ even closer to the MD representation $\rho$, by using the matrix $D_{\tilde{\rho }}(\sigma )$ in (B.29). For an MD representation $\rho$, $D_{\rho }(\sigma )$ is suppose to be signed permutations. So we will try to choose a basis to transform each $D_{\tilde{\rho }}(\sigma )$ into signed permutations. We like to point out that, since both $\tilde{\rho }$ and $\rho$ are symmetric ${\text {SL}}_2({\mathbb {Z}})$ matrix representations that are related by an unitary transformation, according to Theorem 3.4, they can be related by an orthogonal transformation.

Let us consider a simple case to demonstrate our approach. If $\tilde{\rho }(\mathfrak {t})$ is non-degenerate, then $D_{\tilde{\rho }}(\sigma )$ will automatically be a signed permutation matrix. Using signed diagonal matrices $V_\textrm{sd}$, we can transform $\tilde{\rho }$ to many other symmetric representations, $\rho$’s:

$$\begin{aligned} \rho = V_\textrm{sd} \tilde{\rho }V_\textrm{sd} , \end{aligned}$$

(C.1)

where $D_{\rho }(\sigma )$ remains a signed permutation. In fact the signed diagonal matrices $V_\textrm{sd}$ are the most general orthogonal matrices that fix $\tilde{\rho }(\mathfrak {t})$ and transform all $D_{\tilde{\rho }}(\sigma )$’s into (potentially different) signed permutations. Thus the resulting symmetric representations, $\rho$’s, include all the symmetric representations where $D_{\rho }(\sigma )$’s are signed permutations. From those $\rho$’s, we can then construct many pairs of S, T matrices via (3.7), and check which one satisfies the conditions in Proposition B.1. Those S, T matrices that satisfy those conditions may very likely correspond to modular data (or MTC’s). If none of the S, T matrices satisfy the conditions, then the representation $\tilde{\rho }$ will not be an ${\text {SL}}_2({\mathbb {Z}})$ representation of any modular data.

When some eigenspaces of $\tilde{\rho }(\mathfrak {t})$ are more than 1-dimensional, then the $D_{\tilde{\rho }}(\sigma )$ may not be signed permutations. There may be infinite many orthogonal matrices that can transform $D_{\tilde{\rho }}(\sigma )$ into signed permutations, which make the subsequent selection difficult. In the following, we will generalize the above notion of non-degenerate representation, to include some cases where some eigenspaces of $\tilde{\rho }(\mathfrak {t})$ are 2-dimensional or more. We will show that, for those special representations, there is only a finite number of orthogonal matrices that can transform $D_{\tilde{\rho }}(\sigma )$ into signed permutations.

To carry through this program, let us concentrate on an eigenspace $E_{\tilde{\theta }}$ of $\tilde{\rho }(\mathfrak {t})$ corresponding to an eigenvalue $\tilde{\theta }$, and let

$$\begin{aligned} \Omega _{\tilde{\rho }}(\tilde{\theta }) = \{ \sigma \in {\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\tilde{\rho }(\mathfrak {t}))}) \mid \sigma ^2(\tilde{\theta }) =\tilde{\theta }\} \,. \end{aligned}$$

(C.2)

Then $\Omega _{\tilde{\rho }}(\tilde{\theta })$ is a subgroup of ${\text{ Gal }}({\mathbb {Q}}_{{\text {ord}}(\tilde{\rho }(\mathfrak {t}))})$. By definition, $D_{\tilde{\rho }}(\sigma )$ stabilizes the $\tilde{\theta }$-eigenspace $E_{\tilde{\theta }}$ for $\sigma \in \Omega _{\tilde{\rho }}(\tilde{\theta })$, and commute with each other. In particular, $D_{\tilde{\rho }}|_{E_{\tilde{\theta }}}$ (restricted on $E_{\tilde{\theta }}$) defines a representation of $\Omega _{\tilde{\rho }}(\tilde{\theta })$ on $E_{\tilde{\theta }}$.

We can diagonalize $\{D_{\tilde{\rho }}(\sigma )|_{E_{\tilde{\theta }}} \mid \sigma \in \Omega _{\tilde{\rho }}(\tilde{\theta }) \}$ simultaneously within $E_{\tilde{\theta }}$. The degeneracy of the $\tilde{\theta }$-eigenspace $E_{\tilde{\theta }}$ is fully resolved by these $D_{\tilde{\rho }}(\sigma )$’s, if the common eigenspace of these $D_{\tilde{\rho }}(\sigma )|_{E_{\tilde{\theta }}}$’s are all 1-dimensional. In terms of the characters of $\Omega _{\tilde{\rho }}(\tilde{\theta })$, the degeneracy of $E_{\tilde{\theta }}$ can be fully resolved if each irreducible character of $\Omega _{\tilde{\rho }}(\tilde{\theta })$ has multiplicity at most 1 in the character decomposition of $E_{\tilde{\theta }}$ as a representation of $\Omega _{\tilde{\rho }}(\tilde{\theta })$. Now we can introduce the notion of resolved representation:

Definition C.1

A general ${\text {SL}}_2({\mathbb {Z}})$ matrix representation $\tilde{\rho }$ is called resolved if the degeneracy of each of eigenspace of $\tilde{\rho }(\mathfrak {t})$ is fully resolved by $D_{\tilde{\rho }}(\sigma )$, $\sigma \in \Omega _{\tilde{\rho }}(\tilde{\theta })$, as described above.

Given a symmetric irrep-sum matrix representation (denoted as $\rho _\textrm{isum}$), we can use unitary matrices, U’s, to transform it into a symmetric representation $\rho$ via

$$\begin{aligned} \rho (\mathfrak {t}) = U \rho _\textrm{isum}(\mathfrak {t}) U^\dag ,\ \ \ \ \rho (\mathfrak {s}) = U \rho _\textrm{isum}(\mathfrak {s}) U^\dag . \end{aligned}$$

(C.3)

where $D_{\rho }(\sigma )|_{E_{\tilde{\theta }}}$, for all $\sigma \in \Omega _{\tilde{\rho }}(\tilde{\theta })$, are signed permutations within the $\tilde{\theta }$-eigenspace. If $\rho _\textrm{isum}$ is resolved, then there is only a finite number of such representations. We then can check which of those representations satisfy Proposition B.1. This is how we compute the potential modular data S, T’s from resolved $\rho _\textrm{isum}$’s.

To show a resolved $\rho _\textrm{isum}$ is unitarily equivalent to only a finite number representations whose $D_{\rho }(\sigma )|_{E_{\tilde{\theta }}}$ are signed permutations, we note that both $\rho$ and $\rho _\textrm{isum}$ are symmetric, and according to Theorem 3.4, $\rho$ and $\rho _\textrm{isum}$ are in fact orthogonally equivalent, i.e. the above U can be chosen to satisfy $U = U^*$ and $UU^\top ={\text {id}}$. If the number of most general orthogonal matrices U that transform $\rho _\textrm{isum}$ to $\rho$ is finite, then the number of representations $\rho$ are finite.

Since the orthogonal U acts within the eigenspace of $\rho _\textrm{isum}(\mathfrak {t})$, to show the number of possible U’s are finite, we can concentrate on a single $\tilde{\theta }$-eigenspace $E_{\tilde{\theta }}$, and denote $\sigma \in \Omega _{\tilde{\rho }}(\tilde{\theta })$ as $\sigma _\textrm{inv}$. In the following, we will consider the cases where $E_{\tilde{\theta }}$ is 1-dimensional, 2-dimensional, etc.. For each case, we will show the number of possible U’s are finite, and give the possible choices of U’s.

1.1.1 Within an 1-dimensional eigenspace of $\rho _\textrm{isum}(\mathfrak {t})$

$D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}} = \pm 1$ are already signed permutations. In this case the orthogonal matrix U (within the 1-dimensional eigenspace) has only two choices

$$\begin{aligned} U=\pm 1, \end{aligned}$$

(C.4)

which is finite.

1.1.2 Within a 2-dimensional eigenspace of $\rho _\textrm{isum}(\mathfrak {t})$

In this case, the matrix groups MG generated by 2-by-2 matrices, $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$, can have several different forms, for those passing representations. By examine the computer results, we find that, for unresolved cases, matrix groups MG can be

$$\begin{aligned} MG&=\Big \{ \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix} \Big \},{} & {} \text { for } \dim (\rho _\textrm{isum}) \geqslant 5; \nonumber \\ MG&=\Big \{ \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix}, -\begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix} \Big \},{} & {} \text { for } \dim (\rho _\textrm{isum}) \geqslant 6. \end{aligned}$$

(C.5)

For resolved cases, we have

$$\begin{aligned} MG&=\Big \{ \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix}, \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad -1\\ \end{pmatrix} \Big \},{} & {} \text { for } \dim (\rho _\textrm{isum}) \geqslant 4; \nonumber \\ MG&=\Big \{ \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix}, -\begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix}, \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad -1\\ \end{pmatrix}, \begin{pmatrix} -1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix} \Big \},{} & {} \text { for } \dim (\rho _\textrm{isum}) \geqslant 6. \end{aligned}$$

(C.6)

In those two cases,

$$\begin{aligned} U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 1\\ 1&{}\quad -1\\ \end{pmatrix} \ \ \text { or } \ \ U = \frac{1}{\sqrt{2}} \begin{pmatrix} -1&{}\quad 1\\ 1&{}\quad 1\\ \end{pmatrix} \ \ \text { or } \ \ U = \begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad 1\\ \end{pmatrix} \end{aligned}$$

(C.7)

will transform all $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s into signed permutations. In general we have

Theorem C.2

Let

$$\begin{aligned} MG_2&=\Big \{ \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix}, \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad -1\\ \end{pmatrix} \Big \}, \nonumber \\ MG_4&=\Big \{ \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix}, -\begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix}, \begin{pmatrix} 1 &{}\quad 0\\ 0 &{}\quad -1\\ \end{pmatrix}, \begin{pmatrix} -1 &{}\quad 0\\ 0 &{}\quad 1\\ \end{pmatrix} \Big \}. \end{aligned}$$

(C.8)

The most general orthogonal matrices that transform all matrices in $MG_2$ or $MG_4$ into signed permutations must have one of the following forms

$$\begin{aligned} U = \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 1\\ 1&{}\quad -1\\ \end{pmatrix} , \text { or } U = \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} -1&{}\quad 1\\ 1&{}\quad 1\\ \end{pmatrix} , \text { or } U = P V_\textrm{sd} \begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad 1\\ \end{pmatrix} \end{aligned}$$

(C.9)

where $V_\textrm{sd}$ are signed diagonal matrices, and P are permutation matrices. The number of the orthogonal transformations U is finite.

Proof of Theorem C.2

We only needs to consider the first matrix group $MG_2$, where the matrix group is isomorphic to the $\mathbb {Z}_2$ group. There are only four matrix groups formed by 2-dimensional signed permutations matrices, that are isomorphic $\mathbb {Z}_2$. The four matrix groups are generated by the following four generators respectively:

$$\begin{aligned} \begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad -1\\ \end{pmatrix},\ \ \begin{pmatrix} -1&{}\quad 0\\ 0&{}\quad 1\\ \end{pmatrix},\ \ \begin{pmatrix} 0&{}\quad 1\\ 1&{}\quad 0\\ \end{pmatrix},\ \ \begin{pmatrix} 0&{}\quad -1\\ -1&{}\quad 0\\ \end{pmatrix}. \end{aligned}$$

(C.10)

An orthogonal transformation U that transforms MG to one of the above matrix groups must have a from $U=VU_0$, where V transforms $MG_2$ into itself, and $U_0$ is a fixed orthogonal transformation that transforms $MG_2$ to one of the above matrix groups. We can choose $U_0$ to have the following form

$$\begin{aligned} U_0 = \frac{P}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 1\\ 1&{}\quad -1\\ \end{pmatrix} , \text { or } U_0 = \frac{P}{\sqrt{2}} \begin{pmatrix} -1&{}\quad 1\\ 1&{}\quad 1\\ \end{pmatrix} , \text { or } U_0 = P \begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad 1\\ \end{pmatrix}. \end{aligned}$$

(C.11)

To keep MG unchanged V must satisfy

$$\begin{aligned} V \begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad -1\\ \end{pmatrix} = \begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad -1\\ \end{pmatrix} V. \end{aligned}$$

(C.12)

We find that V must be diagonal. Thus V, as an orthogonal matrix, must be signed diagonal. This gives us the result (C.9). $\square$

If $\dim (\rho _\textrm{isum}) \geqslant 8$, it is possible that the matrix group of $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s is generated by the following non-diagonal matrix

$$\begin{aligned} \pm \begin{pmatrix} 0 &{}\quad -1\\ 1 &{}\quad 0\\ \end{pmatrix} \end{aligned}$$

(C.13)

This is because the direct sum decomposition of $\rho _\textrm{isum}$ contains a dimension-6 irreducible representation $6_1^{0,1}$ in “Appendix A”, whose $\rho (\mathfrak {t})$ has a 2-dimensional eigenspace. The representation $6_1^{0,1}$ can give rise to such form of $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s.

The eigenvalues of the matrices are $(\textrm{i},-\textrm{i})$. The most general orthogonal matrices that transform all $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s into signed permutations must have a form

$$\begin{aligned} U = P V_\textrm{sd} \begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad 1\\ \end{pmatrix} . \end{aligned}$$

(C.14)

If $\dim (\rho _\textrm{isum}) \geqslant 8$, it is also possible that $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s form the following matrix group:

$$\begin{aligned} \begin{pmatrix} 1&{}\quad 0 \\ 0 &{}\quad 1 \\ \end{pmatrix} ,\ \ \begin{pmatrix} -\frac{1}{2}&{}\quad -\sqrt{\frac{3}{4}} \\ \sqrt{\frac{3}{4}}&{}\quad -\frac{1}{2} \\ \end{pmatrix} ,\ \ \begin{pmatrix} -\frac{1}{2}&{}\quad \sqrt{\frac{3}{4}} \\ -\sqrt{\frac{3}{4}}&{}\quad -\frac{1}{2} \\ \end{pmatrix} \end{aligned}$$

(C.15)

This is because the direct sum decomposition of $\rho _\textrm{isum}$ contains a dimension-8 irreducible representation whose $\rho (\mathfrak {t})$ has a 2-dimensional eigenspace, which gives rise to the such form of $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s.

The eigenvalues of the later two matrices are $\pm (\textrm{e}^{\textrm{i}2\pi /3},\textrm{e}^{-\textrm{i}2\pi /3})$. A permutation of two elements can only have orders 1 or 2. The corresponding $2 \times 2$ signed permutation matrix can only have eigenvalues 1, $-1$ or $\pm \textrm{i}$. Any other eigenvalue is not possible. Thus, there is no orthogonal matrix that can transform the above two matrices into signed permutation. Such $\rho _\textrm{isum}$ is not a representation of any modular data.

1.1.3 Within a 3-dimensional eigenspace of $\rho _\textrm{isum}(\mathfrak {t})$ for rank $\leqslant 6$

There is only one such case for rank $\leqslant 6$. The $3\times 3$ matrix group MG generated by $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s is given by

$$\begin{aligned} MG&=\Big \{ \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ \end{pmatrix}, \begin{pmatrix} -1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad -1 &{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ \end{pmatrix}, \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad -1 &{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ \end{pmatrix}, \begin{pmatrix} -1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0\\ 0&{}\quad 0&{}\quad 1\\ \end{pmatrix} \Big \}, \nonumber \\ {}&\text { for } \dim (\rho _\textrm{isum}) = 6. \end{aligned}$$

(C.16)

which is resolved. To find the most general orthogonal matrices that transform the above $3\times 3$ matrices in MG into signed permutation matrices, we first show

Theorem C.3

If P is a permutation matrix with $P^2 = {\text {id}}$, then P is a direct sum of $2\times 2$ and $1\times 1$ matrices. If $P_\textrm{sgn}$ is a signed permutation matrix with $P_\textrm{sgn}^2 = {\text {id}}$, then $P_\textrm{sgn}$ is a direct sum of $2\times 2$ and $1\times 1$ matrices.

Proof of Theorem C.3

If P is a permutation matrix with $P^2 = {\text {id}}$, then P must be a pair-wise permutation, and thus P is a direct sum of $2\times 2$ and $1\times 1$ matrices. The reduction from signed permutation matrix to permutation matrix by ignoring the signs is homomorphism of the matrix product. If $P_\textrm{sgn}$ is a signed permutation matrix with $P_\textrm{sgn}^2 = {\text {id}}$, then its reduction gives rise to a permutation matrix P with $P^2 = {\text {id}}$. Since P is a direct sum of $2\times 2$ and $1\times 1$ matrices, $P_\textrm{sgn}$ is also a direct sum of $2\times 2$ and $1\times 1$ matrices. $\square$

Using the above result, similarly, we can show that the most general orthogonal matrices that transform all $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$’s into signed permutations must have a form

$$\begin{aligned} U&= \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 1 &{}\quad 0\\ 1&{}\quad -1 &{}\quad 0\\ 0&{}\quad 0 &{}\quad 1\\ \end{pmatrix} , \text { or } \ U = \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} -1&{}\quad 1 &{}\quad 0\\ 1&{}\quad 1 &{}\quad 0\\ 0&{}\quad 0 &{}\quad 1\\ \end{pmatrix} , \nonumber \\ \text {or }\ U&= \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 0&{}\quad 1 \\ 0&{}\quad 1&{}\quad 0\\ 1&{}\quad 0&{}\quad -1 \\ \end{pmatrix} , \text { or } \ U = \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} -1&{}\quad 0&{}\quad 1 \\ 0&{}\quad 1&{}\quad 0\\ 1&{}\quad 0&{}\quad 1 \\ \end{pmatrix} , \nonumber \\ \text {or }\ U&= \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{}\quad 1\\ 0&{}\quad 1&{}\quad -1 \\ \end{pmatrix} , \text { or } \ U = \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad -1&{}\quad 1\\ 0&{}\quad 1&{}\quad 1 \\ \end{pmatrix} , \nonumber \\ \text {or }\ U&= \frac{PV_\textrm{sd}}{\sqrt{2}} \begin{pmatrix} 1&{}\quad 0&{}\quad 0 \\ 0&{}\quad 1&{}\quad 0\\ 0&{}\quad 0&{}\quad 1 \\ \end{pmatrix} . \end{aligned}$$

(C.17)

where $V_\textrm{sd}$ are signed diagonal matrices, and P are permutation matrices. We note that the non-trivial part of U is a $2\times 2$ block for index (1, 2), (1, 3), and (2, 3). The $2\times 2$ block has three possibilities given in (C.9). Such U’s transform the diagonal matrices in MG into a direct sum of a $2\times 2$ and an $1\times 1$ matrices. This is a general pattern that apply for all resolved diagonal matrix group MG generated by $D_{ \rho _\textrm{isum}}(\sigma _\textrm{inv})|_{E_{\tilde{\theta }}}$.

The above are all the possibilities that can appear in resolved dimension-6 representations. In the following, we will consider more possibilities, that appear only for resolved representations with dimension larger than 6.

1.2 List of S, T matrices from resolved representations

We have constructed a list of irrep-sum symmetric representations (see “Appendix B.2”) that include all the representations of modular data. Among them, we can select a sublist of resolved symmetric representations, denoted as $\{ \rho _\textrm{res} \}$. We then use the orthogonal matrix U constructed above (see (C.4), (C.9) and (C.17)) to transform the resolved symmetric representations $\rho _\textrm{res}$ to representations, $\rho$’s:

$$\begin{aligned} \rho (\mathfrak {t}) = U\rho _\textrm{res}(\mathfrak {t}) U^\top , \ \ \ \rho (\mathfrak {s}) = U\rho _\textrm{res}(\mathfrak {s}) U^\top . \end{aligned}$$

(C.18)

such that the corresponding $D_{\rho }(\sigma )$ are either zero or signed permutation in each eigenspace of $\rho (\mathfrak {t})$. Since the number of such representations is finite, we can examine all resulting representations one by one.

For each U, the resulting representation $\rho$ should satisfy Proposition B.1. In particular, we examine all possible choices of index u that may correspond to the unit object, to see if $\rho$ satisfy the condition (B.9). If no choices of u can satisfy (B.9), then the representations $\rho$ is rejected. If some u’s satisfy (B.9), then for each u, we can construct S, T matrices via (3.7). We then check if the resulting S, T matrices satisfy the conditions of modular data summarized in Proposition B.1

In the following, we list all the pairs of S, T matrices that satisfy the conditions in Proposition B.1, and come from the dimension-6 resolved ${\text {SL}}_2({\mathbb {Z}})$ representations listed in “Appendix B.2”. The list includes all the modular data with $D^2 \notin {\mathbb {Z}}$ from resolved ${\text {SL}}_2({\mathbb {Z}})$ representations (and the list also includes some modular data with $D^2 \in {\mathbb {Z}}$). In the list, the S, T matrices are grouped into orbits generated by Galois conjugations, which are called Galois orbits. To save space, we only list one representative for each orbit. If possible, the representative is chosen to have all-positive quantum dimensions.

Each pair of S, T matrices is indexed by $(r_1, r_2, \cdots ; l_1,l_2,\cdots )_k^a$, such as $(3, 3; 5,4)_2^1$. The first part of index, (3, 3; 5, 4) = (dims;levels), is the index of GT orbit listed in “Appendix B.2”, indicating that the S, T matrices arise from a particular ${\text {SL}}_2({\mathbb {Z}})$ representation in the GT orbit. The subscript k labels the different Galois orbits. The a-index labels the Galois conjugation $\sigma _a: \textrm{e}^{\textrm{i}2\pi /{\text {ord}}(T)} \rightarrow \textrm{e}^{a \textrm{i}2\pi /{\text {ord}}(T)}$. Those a-indexed S, T matrices form a Galois orbit.

Some Galois orbits contain no unitary S, T matrices, but some of those S, T matrices are pseudo-unitary, i.e. those S, T matrices can be obtained from unitary S, T matrices via a change of spherical structure. In this case those Galois orbits can be obtained from Galois orbits that contain Galois orbits. To save space further, we also drop those Galois orbits that contain pseudo-unitary S, T matrices. There is only one orbit which contains no unitary and no pseudo-unitary S, T matrices. The numbering in the following list includes gaps as we maintain the numbering from the arXiv version for consistency.

In the list, T is presented in terms of topological spin $(s_{1},s_{2},\cdots )$ with $s_{i} = \arg (T_{ii})$. S is presented as $(S_{00},S_{01}, S_{02}, S_{03}, \cdots ;\ S_{11}, S_{12}, S_{13}, \cdots )$. $d_i = S_{0i}$ are the quantum dimensions.

Our calculation actually produces 174 pairs of S, T matrices, which are given in Supplementary Material Section in the arXiv version. All those 174 pairs of S, T matrices can be obtained from the pairs of S, T matrices in the following list, via Galois conjugations and change of the spherical structures.

• 1. ind = $(3, 3;5, 4 )_{1}^{1}$: $d_i$ = (1.0, 1.0, 2.0, 2.0, 2.236, 2.236) $D^2=$ 20.0 = 20 $T = ( 0, 0, \frac{1}{5}, \frac{4}{5}, \frac{1}{4}, \frac{3}{4} )$, S = (1, 1, 2, 2, $\sqrt{5}$, $\sqrt{5}$; 1, 2, 2, $-\sqrt{5}$, $-\sqrt{5}$; $-1-\sqrt{5}$, $-1+\sqrt{5}$, 0, 0; $-1-\sqrt{5}$, 0, 0; $-\sqrt{5}$, $\sqrt{5}$; $-\sqrt{5}$)

• 2. ind = $(3, 3;5, 4 )_{2}^{1}$: $d_i$ = (1.0, 1.0, 2.0, 2.0, 2.236, 2.236) $D^2=$ 20.0 = 20 $T = ( 0, 0, \frac{2}{5}, \frac{3}{5}, \frac{1}{4}, \frac{3}{4} )$, S = (1, 1, 2, 2, $\sqrt{5}$, $\sqrt{5}$; 1, 2, 2, $-\sqrt{5}$, $-\sqrt{5}$; $-1+\sqrt{5}$, $-1-\sqrt{5}$, 0, 0; $-1+\sqrt{5}$, 0, 0; $\sqrt{5}$, $-\sqrt{5}$; $\sqrt{5}$)

• 3. ind = $(4, 2;15, 5 )_{1}^{1}$: $d_i$ = (1.0, 1.0, 1.0, 1.618, 1.618, 1.618) $D^2=$ 10.854 = $\frac{15+3\sqrt{5}}{2}$ $T = ( 0, \frac{1}{3}, \frac{1}{3}, \frac{2}{5}, \frac{11}{15}, \frac{11}{15} )$, S = (1, 1, 1, $\frac{1+\sqrt{5}}{2}$, $\frac{1+\sqrt{5}}{2}$, $\frac{1+\sqrt{5}}{2}$; $\zeta _{3}^{1}$, $-\zeta _{6}^{1}$, $\frac{1+\sqrt{5}}{2}$, $\frac{1+\sqrt{5}}{2}\zeta _{3}^{1}$, $-\frac{1+\sqrt{5}}{2}\zeta _{6}^{1}$; $\zeta _{3}^{1}$, $\frac{1+\sqrt{5}}{2}$, $-\frac{1+\sqrt{5}}{2}\zeta _{6}^{1}$, $\frac{1+\sqrt{5}}{2}\zeta _{3}^{1}$; $-1$, $-1$, $-1$; $-\zeta _{3}^{1}$, $\zeta _{6}^{1}$; $-\zeta _{3}^{1}$)

• 7. ind = $(4, 2;7, 3 )_{1}^{1}$: $d_i$ = (1.0, 3.791, 3.791, 3.791, 4.791, 5.791) $D^2=$ 100.617 = $\frac{105+21\sqrt{21}}{2}$ $T = ( 0, \frac{1}{7}, \frac{2}{7}, \frac{4}{7}, 0, \frac{2}{3} )$, S = (1, $\frac{3+\sqrt{21}}{2}$, $\frac{3+\sqrt{21}}{2}$, $\frac{3+\sqrt{21}}{2}$, $\frac{5+\sqrt{21}}{2}$, $\frac{7+\sqrt{21}}{2}$; $2-c^{1}_{21} -2c^{2}_{21} +3c^{3}_{21} +2c^{4}_{21} -2c^{5}_{21}$, $-c^{2}_{21} -2c^{3}_{21} -c^{4}_{21} +c^{5}_{21}$, $-1+2c^{1}_{21} +3c^{2}_{21} -c^{3}_{21} +2c^{5}_{21}$, $-\frac{3+\sqrt{21}}{2}$, 0; $-1+2c^{1}_{21} +3c^{2}_{21} -c^{3}_{21} +2c^{5}_{21}$, $2-c^{1}_{21} -2c^{2}_{21} +3c^{3}_{21} +2c^{4}_{21} -2c^{5}_{21}$, $-\frac{3+\sqrt{21}}{2}$, 0; $-c^{2}_{21} -2c^{3}_{21} -c^{4}_{21} +c^{5}_{21}$, $-\frac{3+\sqrt{21}}{2}$, 0; 1, $\frac{7+\sqrt{21}}{2}$; $-\frac{7+\sqrt{21}}{2}$)

• 9. ind = $(6;9 )_{1}^{1}$: $d_i$ = (1.0, 0.347, 1.0, 1.532, $-1.0$, $-1.879$) $D^2=$ 9.0 = 9 $T = ( 0, \frac{1}{9}, \frac{2}{3}, \frac{4}{9}, \frac{1}{3}, \frac{7}{9} )$, S = (1, $c^{2}_{9}$, 1, $c^{1}_{9}$, $-1$, $c_9^4$; 1, $c^{1}_{9}$, 1, $-c_9^4$, 1; 1, $c_9^4$, $-1$, $c^{2}_{9}$; 1, $-c^{2}_{9}$, 1; 1, $-c^{1}_{9}$; 1)

• 10. ind = $(6;13 )_{1}^{1}$: $d_i$ = (1.0, 1.941, 2.770, 3.438, 3.907, 4.148) $D^2=$ 56.746 = $21+15c^{1}_{13} +10c^{2}_{13} +6c^{3}_{13} +3c^{4}_{13} +c^{5}_{13}$ $T = ( 0, \frac{4}{13}, \frac{2}{13}, \frac{7}{13}, \frac{6}{13}, \frac{12}{13} )$, S = (1, $\xi _{13}^{2}$, $\xi _{13}^{3}$, $\xi _{13}^{4}$, $\xi _{13}^{5}$, $\xi _{13}^{6}$; $-\xi _{13}^{4}$, $\xi _{13}^{6}$, $-\xi _{13}^{5}$, $\xi _{13}^{3}$, $-1$; $\xi _{13}^{4}$, 1, $-\xi _{13}^{2}$, $-\xi _{13}^{5}$; $\xi _{13}^{3}$, $-\xi _{13}^{6}$, $\xi _{13}^{2}$; $-1$, $\xi _{13}^{4}$; $-\xi _{13}^{3}$)

• 12. ind = $(6;16 )_{1}^{1}$: $d_i$ = (1.0, 1.0, 1.0, 1.0, 1.414, 1.414) $D^2=$ 8.0 = 8 $T = ( 0, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{1}{16}, \frac{5}{16} )$, S = (1, 1, 1, 1, $\sqrt{2}$, $\sqrt{2}$; 1, 1, 1, $-\sqrt{2}$, $-\sqrt{2}$; $-1$, $-1$, $\sqrt{2}$, $-\sqrt{2}$; $-1$, $-\sqrt{2}$, $\sqrt{2}$; 0, 0; 0)

• 16. ind = $(6;16 )_{2}^{1}$: $d_i$ = (1.0, 1.0, 1.0, 1.0, 1.414, 1.414) $D^2=$ 8.0 = 8 $T = ( 0, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \frac{1}{16}, \frac{13}{16} )$, S = (1, 1, 1, 1, $\sqrt{2}$, $\sqrt{2}$; 1, 1, 1, $-\sqrt{2}$, $-\sqrt{2}$; $-1$, $-1$, $-\sqrt{2}$, $\sqrt{2}$; $-1$, $\sqrt{2}$, $-\sqrt{2}$; 0, 0; 0)

• 20. ind = $(6;35 )_{1}^{1}$: $d_i$ = (1.0, 1.618, 1.801, 2.246, 2.915, 3.635) $D^2=$ 33.632 = $15+3c^{1}_{35} +2c^{4}_{35} +6c^{5}_{35} +3c^{6}_{35} +3c^{7}_{35} +2c^{10}_{35} +2c^{11}_{35}$ $T = ( 0, \frac{2}{5}, \frac{1}{7}, \frac{5}{7}, \frac{19}{35}, \frac{4}{35} )$, S = (1, $\frac{1+\sqrt{5}}{2}$, $\xi _{7}^{2}$, $\xi _{7}^{3}$, $c^{1}_{35} +c^{6}_{35}$, $c^{1}_{35} +c^{4}_{35} +c^{6}_{35} +c^{11}_{35}$; $-1$, $c^{1}_{35} +c^{6}_{35}$, $c^{1}_{35} +c^{4}_{35} +c^{6}_{35} +c^{11}_{35}$, $-\xi _{7}^{2}$, $-\xi _{7}^{3}$; $-\xi _{7}^{3}$, 1, $-c^{1}_{35} -c^{4}_{35} -c^{6}_{35} -c^{11}_{35}$, $\frac{1+\sqrt{5}}{2}$; $-\xi _{7}^{2}$, $\frac{1+\sqrt{5}}{2}$, $-c^{1}_{35} -c^{6}_{35}$; $\xi _{7}^{3}$, $-1$; $\xi _{7}^{2}$)

• 24. ind = $(6;56 )_{1}^{1}$: $d_i$ = (1.0, 1.0, 1.801, 1.801, 2.246, 2.246) $D^2=$ 18.591 = $12+6c^{1}_{7} +2c^{2}_{7}$ $T = ( 0, \frac{1}{4}, \frac{1}{7}, \frac{11}{28}, \frac{5}{7}, \frac{27}{28} )$, S = (1, 1, $\xi _{7}^{2}$, $\xi _{7}^{2}$, $\xi _{7}^{3}$, $\xi _{7}^{3}$; $-1$, $\xi _{7}^{2}$, $-\xi _{7}^{2}$, $\xi _{7}^{3}$, $-\xi _{7}^{3}$; $-\xi _{7}^{3}$, $-\xi _{7}^{3}$, 1, 1; $\xi _{7}^{3}$, 1, $-1$; $-\xi _{7}^{2}$, $-\xi _{7}^{2}$; $\xi _{7}^{2}$)

• 28. ind = $(6;80 )_{1}^{1}$: $d_i$ = (1.0, 1.0, 1.414, 1.618, 1.618, 2.288) $D^2=$ 14.472 = $10+2\sqrt{5}$ $T = ( 0, \frac{1}{2}, \frac{1}{16}, \frac{2}{5}, \frac{9}{10}, \frac{37}{80} )$, S = (1, 1, $\sqrt{2}$, $\frac{1+\sqrt{5}}{2}$, $\frac{1+\sqrt{5}}{2}$, $c^{3}_{40} +c^{5}_{40} -c^{7}_{40}$; 1, $-\sqrt{2}$, $\frac{1+\sqrt{5}}{2}$, $\frac{1+\sqrt{5}}{2}$, $-c^{3}_{40} -c^{5}_{40} +c^{7}_{40}$; 0, $c^{3}_{40} +c^{5}_{40} -c^{7}_{40}$, $-c^{3}_{40} -c^{5}_{40} +c^{7}_{40}$, 0; $-1$, $-1$, $-\sqrt{2}$; $-1$, $\sqrt{2}$; 0)

• 36. ind = $(6;80 )_{2}^{1}$: $d_i$ = (1.0, 1.0, 1.414, 1.618, 1.618, 2.288) $D^2=$ 14.472 = $10+2\sqrt{5}$ $T = ( 0, \frac{1}{2}, \frac{3}{16}, \frac{2}{5}, \frac{9}{10}, \frac{47}{80} )$, S = (1, 1, $\sqrt{2}$, $\frac{1+\sqrt{5}}{2}$, $\frac{1+\sqrt{5}}{2}$, $c^{3}_{40} +c^{5}_{40} -c^{7}_{40}$; 1, $-\sqrt{2}$, $\frac{1+\sqrt{5}}{2}$, $\frac{1+\sqrt{5}}{2}$, $-c^{3}_{40} -c^{5}_{40} +c^{7}_{40}$; 0, $c^{3}_{40} +c^{5}_{40} -c^{7}_{40}$, $-c^{3}_{40} -c^{5}_{40} +c^{7}_{40}$, 0; $-1$, $-1$, $-\sqrt{2}$; $-1$, $\sqrt{2}$; 0)

The above list include all rank-6 modular data with non-integral $D^2$ and coming from resolved ${\text {SL}}_2({\mathbb {Z}})$ representations (as well as some with $D^2$ integral, as we filter using conditions that imply $D^2\in \mathbb {Z}$, but not conversely). The list misses two known modular data with non-integral $D^2 = 74.617$, whose topological spins are $s_i = (0,\frac{1}{9},\frac{1}{9},\frac{1}{9},\frac{1}{3},\frac{2}{3})$ or $s_i= (0,\frac{8}{9},\frac{8}{9},\frac{8}{9},\frac{1}{3},\frac{2}{3})$. From those $s_i$’s, we find that they must come from the unresolved GT orbit (4, 1, 1; 9, 1, 1). In the main text of this paper, we showed that the unresolved ${\text {SL}}_2({\mathbb {Z}})$ representations can only produce such modular data (and its conjugations by Galois action and signed diagonal matrices). The unresolved cases are handled in the main text of the paper, which leads to a complete classification of all rank-6 modular data.