Towards a twistless converse theorem

Abstract

We prove a finiteness theorem for Dirichlet series satisfying the Euler product, analytic continuation, and functional equation expected of a holomorphic newform. For any fixed level $N$, even weight $k$, and exponent $σ$, only finitely many such series have Dirichlet coefficients $\ll n^σ$. No hypotheses on twists by Dirichlet characters are assumed. The proof combines a combinatorial spanning theorem for Ramanujan sums with an operator-theoretic argument that uses two distinct primes to establish discreteness of the possible Hecke eigenvalues.