Linear relations among the Fourier coefficients of modular forms on groups Γ0(N) of genus zero and their applications

We study the linear relations among the Fourier coefficients of modular forms on the group Γ0(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 ΔN(z) of weight 12 on Γ0(N) with a zero of maximum order at ∞ is listed.