Modular forms on Hecke’s modular groups

Abstract:

Let $H = \{ \tau = x + iy:y > 0\}$. Let $\lambda > 0,k > 0,\gamma = \pm 1$. Let $M(\lambda ,k,\gamma )$ denote the set of functions f for which $f(\tau ) = \sum _{n = 0}^\infty {a_n}{e^{2\pi in\tau /\lambda }}$ and $f( - 1/\tau ) = \gamma {(\tau /i)^k}f(\tau )$, for all $\tau \in H$. Let ${M_0}(\lambda ,k,\gamma )$ denote the set of $f \in M(\lambda ,k.\gamma )$ for which $f(\tau ) = O({y^c})$ uniformly for all x as $y \to {0^ + }$, for some real c. We give a new proof that if $\lambda = 2\cos (\pi /q)$ for an integer $q \geqq 3$, then $M(\lambda ,k,\gamma ) = {M_0}(\lambda ,k,\gamma )$.