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 $F(s)$.

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