Expressions for weight 2 cusp forms in holomorphic eta quotients

TL;DR

This paper explores expressing weight 2 cusp forms as eta quotients for levels up to 100, finding explicit formulas for most cases and developing methods to minimize terms and analyze zeros.

Contribution

It provides explicit eta quotient expressions for nearly all weight 2 cusp forms up to level 100 and introduces techniques to find concise formulas and study zeros.

Findings

Expressions found for all but 4 forms

Methods to minimize terms in eta quotient expressions

Applications to zeroes of modular forms

Abstract

We attempt to compute expressions in terms of the Dedekind eta function for all weight 2 new cusp forms with level up to 100, using methods of Allen et. al. In cases where no expression exists, we raise the level instead of the weight, meaning our eta quotients are always holomorphic. Of the forms we examine, we find expressions for all but 4. We also present methods to find expressions with relatively few terms, and how these expressions can be used to demonstrate zeroes of modular forms.