Theta Products and Eta Quotients of Level $24$ and Weight $2$

TL;DR

This paper constructs bases for certain modular form spaces of level 24 and weight 2, computes Fourier coefficients of theta products, and derives formulas for counting representations of integers by specific quadratic forms, including Ramanujan's universal forms.

Contribution

It provides explicit bases, Fourier coefficients, and representation formulas for quadratic forms in modular form spaces of level 24, with new uniform methods and complete eta quotient classifications.

Findings

Fourier coefficients for 35 theta products are determined.

Formulas for counting representations by specific quadratic forms are established.

All eta quotients in certain Eisenstein spaces are identified and their coefficients computed.

Abstract

We find bases for the spaces $M_2\Big(\Gamma_0(24),\Big(\frac{d}{\cdot}\Big)\Big)$ ($d=1,8,12, 24$) of modular forms. We determine the Fourier coefficients of all $35$ theta products $\varphi[a_1,a_2,a_3,a_4](z)$ in these spaces. We then deduce formulas for the number of representations of a positive integer $n$ by diagonal quaternary quadratic forms with coefficients $1$, $2$, $3$ or $6$ in a uniform manner, of which $14$ are Ramanujan's universal quaternary quadratic forms. We also find all the eta quotients in the Eisenstein spaces $E_2\Big(\Gamma_0(24),\Big(\frac{d}{\cdot}\Big)\Big)$ ($d=1,8,12,24$) and give their Fourier coefficients.