Zeros of Newform Eisenstein Series on $\Gamma_0(N)$

TL;DR

This paper studies the zeros of newform Eisenstein series on modular groups, providing insights into their location and distribution for large weights, which advances understanding of automorphic forms.

Contribution

It determines the location and distribution of many zeros of newform Eisenstein series on $ ext{Gamma}_0(N)$ for large weights, a novel analysis in this area.

Findings

Significant fraction of zeros located and distributed explicitly

Zeros analyzed for Eisenstein series on $ ext{Gamma}_0(N)$

Results valid for sufficiently large weights

Abstract

We examine the zeros of newform Eisenstein series $E_{\chi_1,\chi_2,k}(z)$ of weight $k$ on $\Gamma_0(q_1 q_2)$, where $\chi_1$ and $\chi_2$ are primitive characters modulo $q_1$ and $q_2$, respectively. We determine the location and distribution of a significant fraction of the zeros of these Eisenstein series for $k$ sufficiently large.