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We study graded rings of modular forms over congruence subgroups, with coefficients in a subring A of C, 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 C-algebras of modular forms over Γ1(N) and Γ0(N) with some conditions on N. For N⩾5, we prove that the Z[1/N]-algebra...
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