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

Abstract

We find bases for the spaces $M_2({\Gamma }_0(24),(\frac{d}{\cdot }\left)\right)$ ($d=1,8,12,24$) of modular forms. We determine the Fourier coefficients of all $35$ theta products $\phi [a_1,a_2,a_3,a_4](z)$ 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 $E_2({\Gamma }_0(24),(\frac{d}{\cdot }\left)\right)$ ($d=1,8,12,24$) and give their Fourier coefficients.

arXiv comment27 pages