For the most part we have considered modular forms on SL2(Z). Incidentally in dealing with Hecke operators on such forms, we needed to pass to congruence subgroups. We want to return more systematically to modular forms on such subgroups. As already mentioned, there are three important such subgroups, which we called Γ0(N), Γ1(N) and Γ(N). We treat Γ1(N) in some detail as an example. By conjugation, one can reduce the theory on Γ(N) to that of Γ1(N).
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