Linear relations among the Fourier coefficients of modular forms on groups [formula omitted] of genus zero and their applications

Abstract

We study the linear relations among the Fourier coefficients of modular forms on the group Γ 0 ( N ) of genus zero. Applying these linear relations, congruence properties of Hecke eigenforms, replicable properties of Hauptmoduln and congruences of representation numbers of the sums of n squares can be obtained. The eta-quotient expression of the unique normalized modular form Δ N ( z ) of weight 12 on Γ 0 ( N ) with a zero of maximum order at ∞ is listed.