2 2 Fe b 19 95 AN EXTENSION OF HECKE ’ S CONVERSE THEOREM

Associated to a newform f (z) is a Dirichlet series L f (s) with functional equation and Euler product. Hecke showed that if the Dirichlet series F (s) has a functional equation of the appropriate form, then F (s) = L f (s) for some holomorphic newform f (z) on Γ(1). Weil extended this result to Γ 0 (N) under an assumption on the twists of F (s) by Dirichlet characters. We show that, at least for small N, the assumption on twists can be replaced by an assumption on the local factors of the Euler product of F (s).