The Values of Modular Functions and Modular Forms

Abstract

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 $\mathfrak{H}\left( \text{here}\,\,{{q}_{\ell }}\,:=\,{{e}^{2\pi iz/\ell }} \right)$. We find universal recursive formulas for $q$-series coefficients of any modular form on $\Gamma _{0}^{+}\left( p \right)$ in terms of those of the canonical Hauptmodul $j_{p}^{+}$.