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