A modular analogue of a problem of Vinogradov

Abstract

Given a primitive, non-CM, holomorphic cusp form f with normalized Fourier coefficients a(n) and given an interval $$I\subset [-2, 2]$$ , we study the least prime p such that $$a(p)\in I$$ . This can be viewed as a modular form analogue of Vinogradov’s problem on the least quadratic non-residue. We obtain strong explicit bounds on p, depending on the analytic conductor of f for some specific choices of I.