Generating spaces of modular forms with $\eta$-quotients

TL;DR

This paper explores which spaces of classical modular forms can be generated by sums of eta-quotients, providing new examples and demonstrating that all modular forms of level (N) can be expressed as rational functions of eta-products.

Contribution

It introduces new examples of modular form spaces generated by eta-quotients and proves that all modular forms at level (N) can be written as rational functions of eta-products.

Findings

New examples of modular form spaces generated by eta-quotients

All modular forms of level (N) can be expressed as rational functions of eta-products

Extended understanding of the structure of modular forms in terms of eta-quotients

Abstract

In this note we consider a question of Ono, concerning which spaces of classical modular forms can be generated by sums of $\eta$-quotients. We give some new examples of spaces of modular forms which can be generated as sums of $\eta$-quotients, and show that we can write all modular forms of level $\Gamma_0(N)$ as rational functions of $\eta$-products.