# Eta-quotients of Prime or Semiprime Level and Elliptic Curves

## TL;DR

This paper explores eta-quotients of prime or semiprime level, their modular properties, and their relation to elliptic curves, providing new examples and an algorithmic approach for representing elliptic curves via eta-quotients.

## Contribution

It demonstrates that eta-quotients modular for certain levels are also modular for (N), classifies when even weight eta-quotients exist for specific groups, and offers new elliptic curve examples with a method to find more.

## Key findings

- Eta-quotients modular for levels coprime to 6 are also modular for (N).
- Classification of even weight eta-quotients in specific modular forms spaces.
- New elliptic curve examples with modular forms as linear combinations of eta-quotients.

## Abstract

From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong. In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for $\Gamma_0(N)$. We then categorize when even weight eta-quotients can exist in $M_k (\Gamma_1(p))$ and $M_k (\Gamma_1(pq))$, for distinct primes $p,q$. We conclude by providing some new examples of elliptic curves whose corresponding modular forms can be written as a linear combination of eta-quotients, and describe an algorithmic method for finding additional examples.