The prime level N=2

Abstract

For a real number λ>0 the Hecke group G(λ) is defined to be the subgroup of SL₂(ℝ) which is generated by $T^{\lambda}= \bigl(\begin{array}{cc}\scriptstyle{1} & \scriptstyle{\lambda}\\[-2mm]\scriptstyle{0} &\scriptstyle{1}\end{array}\bigr)$ and $S = \bigl(\begin{array}{cc}\scriptstyle{0} & \scriptstyle{1} \\[-2mm] \scriptstyle{- 1} &\scriptstyle{0}\end{array}\bigr)$. We mention the Hecke groups not because of Hecke’s pioneering research (Hecke in Lectures on Dirichlet Series, Modular Functions and Quadratic Forms, Vandenhoeck & Ruprecht, Göttingen, 1983), but merely since three of them are conjugate to Fricke groups: Besides the modular group G(1)=Γ₁ itself, we have

$$M G(\sqrt{N}) M^{- 1} = \Gamma^*(N) \hspace{4mm} \mbox{with}\hspace{4mm} M =\left(\begin{array}{cc}1 & 0 \\ 0 &\sqrt{N}\end{array}\right)\quad \mbox{for}\quad N \in\{ 2, 3, 4 \}.$$

The Hecke group G(2) is also called the theta group since Jacobi’s θ(z) is a modular form for G(2). Several of the results in Sects. 10, 11 and 13 are transcriptions of earlier research (Köhler in Abh. Math. Sem. Univ. Hamburg 55, 75–89, 1985), (Köhler in Math. Z. 197, 69–96, 1988), (Köhler in Abh. Math. Sem. Univ. Hamburg 58, 15–45, 1988) on theta series on these three Hecke groups.