The average least negative Hecke eigenvalue

Abstract

We consider the average point at which the Fourier coefficients of newforms first
change sign. The first sign change is referred to as the least negative Hecke eigenvalue, and we show that it has a finite average. We consider this average in the prime case and the general case.

In exploring this problem, we use large sieve techniques for cusp forms. We consider two different approaches to finding large sieve inequalities, with each approach giving better results in certain ranges.

We also look into the distribution of eigenvalues at primes that divide the level,
as these eigenvalues have a different arithmetic structure to eigenvalues at primes that do not divide the level. This is explored through the trace of Hecke operators.

Date of Award	20 Jan 2026
Original language	English
Awarding Institution	University of Bristol
Supervisor	Jonathan W Bober (Supervisor)