On the average least negative Hecke eigenvalue

TL;DR

This paper investigates the average position of the first negative Hecke eigenvalue in classical newforms, providing explicit computations and drawing analogies to the least quadratic non-residue problem, under GRH.

Contribution

It introduces a method to compute the average least negative Hecke eigenvalue and distinguishes between prime and general cases, extending large sieve inequalities.

Findings

Average least negative prime Hecke eigenvalue matches the least quadratic non-residue under GRH.

Develops uniform large sieve inequalities in weight and level aspects.

Provides explicit finite mean and its computation for the first sign change.

Abstract

We show that the first sign change of Hecke eigenvalues of classical newforms has a finite mean, which we also compute. We distinguish between the first negative prime Hecke eigenvalue, and the first negative Hecke eigenvalue. This problem can be considered to be an analogue of the least quadratic non-residue problem, of which the average was explored by Erd\H{o}s in 1961. In fact, the average least negative prime Hecke eigenvalue has the same value as the average least quadratic non-residue, under GRH. To compute these averages, we develop large sieve inequalities that are uniform in both the weight and level aspect.