Distinguishing newforms by their Hecke eigenvalues

Abstract

Let f, g be distinct normalized Hecke eigenforms of weights $k_1,k_2$ lying in the subspace of newforms with Fourier coefficients $\{ n^{(k_1 -1)/2} \lambda _f(n)\}_{n \in \mathbb {N}}$ and $\{ n^{(k_2 -1)/2}\lambda _g(n) \}_{ n \in \mathbb {N}}$ respectively. For such newforms f, g of CM type and primes p, we study the natural density of the set

$$\begin{aligned} S = \{ p ~|~ \lambda _f(p) = \lambda _g(p) \}. \end{aligned}$$

We show that the upper natural density of S is $\le 3/4$ if $f \ne g$ and it is equal to 1/2 when f and g have different weights and have the same associated CM (quadratic) field. Further, f and g have different associated CM (quadratic) fields if and only if the natural density of S is 1/4. When at least one of f, g is a non-CM form, we study the natural density of the sets

$$\begin{aligned} S_{+}(x, \alpha ) = \{ p \le x ~|~ \theta _f(p) + \theta _g(p) = \alpha \} \text { and }\\ S_{-}(x, \beta ) = \{ p \le x ~|~ \theta _f(p) - \theta _g(p) = \beta \} \end{aligned}$$

where $\theta _f(p), \theta _g(p) \in [0, \pi ]$ are the angles associated to the p-th Hecke eigen values of f, g respectively and $\alpha \in [0, 2\pi ], ~\beta \in [-\pi , \pi ]$. In this case, we show that $S_{+}(x, \alpha )$ and $S_{-}(x, \beta )$ have natural density zero when f and g are distinct and not twists of each other. Finally, we establish an explicit link between the elements of these sets and sign changes of Fourier coefficients at prime powers which allows us to improve a number of existing results in this set-up.