The Values of Modular Functions and Modular Forms

Abstract

AbstractLet Γ0 be a Fuchsian group of the first kind of genus zero and Γ be a subgroup of Γ0 of finite index of genus zero. We find universal recursive relations giving the qr-series coefficients of j0 by using those of the qhs -series of j, where j is the canonical Hauptmodul for Γ and j0 is a Hauptmodul for Γ0 without zeros on the complex upper half plane (here qℓ := e2πiz/ℓ). We find universal recursive formulas for q-series coefficients of any modular form on in terms of those of the canonical Hauptmodul .