A simple method to extract the zeros of some Eisenstein series of level $N$

TL;DR

This paper introduces a straightforward approach to determine the zeros of certain weight two Eisenstein series of levels 2, 3, 5, and 7, leveraging their algebraic properties and relation to classical modular forms.

Contribution

It presents a novel, simple method for extracting zeros of specific Eisenstein series based on their algebraic structure and connection to classical modular forms.

Findings

Zeros are controlled by those of E_4 and E_6 in the fundamental domain.

Method applies to Eisenstein series of levels 2, 3, 5, and 7.

Provides explicit zero locations for these series.

Abstract

This paper provides a simple method to extract the zeros of some weight two Eisenstein series of level $N$ where $N=2,3,5$ and $7$. The method is based on the observation that these Eisenstein series are integral over the graded algebra of modular forms on $SL(2,Z)$ and their zeros are `controlled' by those of $E_4$ and $E_6$ in the fundamental domain of $\Gamma_0(N)$.