A modular analogue of a problem of Vinogradov

TL;DR

This paper investigates the smallest prime for which the Fourier coefficient of a non-CM holomorphic cusp form falls within a given interval, providing explicit bounds based on the form's analytic conductor.

Contribution

It introduces a modular form analogue of Vinogradov's problem and derives explicit bounds on the least prime with specified Fourier coefficient properties.

Findings

Established explicit bounds on the least prime p

Connected Fourier coefficients to Vinogradov's classical problem

Provided bounds depending on the analytic conductor

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 $I$.