Generators of graded rings of modular forms

We study graded rings of modular forms over congruence subgroups, with coefficients in a subring A   of CC, and specifically the highest weight needed to generate these rings as A-algebras. In particular, we determine upper bounds, independent of N  , for the highest needed weight that generates the CC-algebras of modular forms over Γ1(N)Γ1(N) and Γ0(N)Γ0(N) with some conditions on N  . For N⩾5N⩾5, we prove that the Z[1/N]Z[1/N]-algebra of modular forms over Γ1(N)Γ1(N) with coefficients in Z[1/N]Z[1/N] is generated in weight at most 3. We give an algorithm that computes the generators, and supply some computations that allow us to state two conjectures concerning the situation over Γ0(N)Γ0(N).