A family of Eta Quotients and an Extension of the Ramanujan-Mordell Theorem

TL;DR

This paper introduces a family of eta quotients forming a basis for certain cusp form spaces and extends the Ramanujan-Mordell formula using these basis elements and modular form properties.

Contribution

It constructs a new basis of eta quotients for cusp form spaces and extends the Ramanujan-Mordell formula leveraging this basis and modular form properties.

Findings

The family _{j,k}(z) forms a basis for S_{2k}(\u03b612(12)).

The paper extends the Ramanujan-Mordell formula using this basis.

Provides explicit eta quotient basis for specific cusp form spaces.

Abstract

Let $k\geq 2$ be an integer and $j$ an integer satisfying $1\leq j \leq 4k-5$. We define a family $\{ C_{j,k}(z) \}_{1\leq j \leq 4k-5} $ of eta quotients, and prove that this family constitute a basis for the space $S_{2k} (\Gamma_0 (12))$ of cusp forms of weight $2k$ and level $12$. We then use this basis together with certain properties of modular forms at their cusps to prove an extension of the Ramanujan-Mordell formula.