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arXiv · 1607.03997

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

Ayşe Alaca·Şaban Alaca·Zafer Selcuk Aygin
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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.

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Ayşe Alaca, Şaban Alaca, Zafer Selcuk Aygin. 2016-07-14. Theta Products and Eta Quotients of Level $24$ and Weight $2$. https://arxiv.org/abs/1607.03997

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