The Values of Modular Functions and Modular Forms

Let Γ 0 be a Fuchsian group of the first kind of genus zero and Γ be a subgroup of Γ 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 hs -series of j , where j is the canonical Hauptmodul for Γ and j 0 is a Hauptmodul for Γ 0 without zeros on the complex upper half plane (here q ℓ := e 2π iz /ℓ ). We find universal recursive formulas for q -series coefficients of any modular form on in terms of those of the canonical Hauptmodul .