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(h5)-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 5 (here q(l) := e(2 pi iz/l)). 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)(+).