The first negative Fourier coefficient of an Eisenstein series newform

TL;DR

This paper corrects and proves a conjecture about the average position of the first negative Fourier coefficient in Eisenstein series newforms, contributing to understanding their sign change behavior.

Contribution

It corrects and proves a conjecture regarding the average index of the first negative Fourier coefficient in Eisenstein series newforms.

Findings

Confirmed the corrected conjecture about sign change distribution.

Established the average position of the first negative Fourier coefficient.

Enhanced understanding of Fourier coefficient sign patterns in Eisenstein series.

Abstract

There have been a number of papers on statistical questions concerning the sign changes of Fourier coefficients of newforms. In one such paper, Linowitz and Thompson gave a conjecture describing when, on average, the first negative sign of the Fourier coefficients of an Eisenstein series newform occurs. In this paper, we correct their conjecture and prove the corrected version.