The Values of Modular Functions and Modular Forms

http://dx.doi.org/10.4153/CMB-2006-050-4
Canad. Math. Bull. 49(2006), 526-535

  Published:2006-12-01
 Printed: Dec 2006

Let $\Gamma_0$ be a Fuchsian group of the first kind of genus zero and $\Gamma$ be a subgroup of $\Gamma_0$ of finite index of genus zero. We find universal recursive relations giving the $q_{r}$-series coefficients of $j_0$ by using those of the $q_{h_{s}}$-series of $j$, where $j$ is the canonical Hauptmodul for $\Gamma$ and $j_0$ is a Hauptmodul for $\Gamma_0$ without zeros on the complex upper half plane $\mathfrak{H}$ (here $q_{\ell} := e^{2 \pi i z / \ell}$). We find universal recursive formulas for $q$-series coefficients of any modular form on $\Gamma_0^{+}(p)$ in terms of those of the canonical Hauptmodul $j_p^{+}$.