Twisting moduli for GL(2)

We prove various converse theorems for automorphic forms on Γ0(N), each assuming fewer twisted functional equations than the last. We show that no twisting at all is needed for holomorphic modular forms in the case that N∈{18, 20, 24}– these integers are the smallest multiples of 4or 9not covered by earlier work of Conrey–Farmer. This development is a consequence of finding generating sets for Γ0(N)such that each generator can be written as a product of special matrices. As for real-analytic Maass forms of even (resp. odd) weight we prove the analogous statement for 1 ≤N≤12and N∈{16, 18}(resp. 1 ≤N≤12, 14 ≤N≤18 and N∈{20, 23, 24}).