A modular analogue of a problem of Vinogradov

Ratnadeep Acharya, Sary Drappeau, Satadal Ganguly et Olivier Ramaré

Published 2022-08-31Version 1

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 $I$.