A note on GL₂ converse theorems

Weil's well-known converse theorem shows that modular forms f ∈ M_k(Γ₀(q)) are characterized by the functional equation for twists of L_f(s). Conrey-Farmer had partial success at replacing the assumption on twists by the assumption of L_f(s) having an Euler product of the appropriate form. In this Note we obtain a hybrid version of Weil's and Conrey-Farmer's results, by proving a converse theorem for all q ≥ 1 under the assumption of the Euler product and, moreover, of the functional equation for the twists to a single modulus.