Functiones et Approximatio Commentarii Mathematici

Theta products and eta quotients of level $24$ and weight $2$

Ayşe Alaca, Şaban Alaca, and Zafer Selcuk Aygin

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We find bases for the spaces $M_2\Big(\Gamma_0(24),(\frac{d}{\cdot})\Big)$ ($d=1,8,12, 24$) 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 $E_2\Big(\Gamma_0(24),(\frac{d}{\cdot})\Big)$ ($d=1,8,12, 24$) and give their Fourier coefficients.

Article information

Funct. Approx. Comment. Math., Volume 57, Number 2 (2017), 205-234.

First available in Project Euclid: 28 March 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 11F20: Dedekind eta function, Dedekind sums 11F27: Theta series; Weil representation; theta correspondences 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 11F30: Fourier coefficients of automorphic forms

Dedekind eta function eta quotients theta products Eisenstein series modular forms cusp forms Fourier coefficients Fourier series


Alaca, Ayşe; Alaca, Şaban; Aygin, Zafer Selcuk. Theta products and eta quotients of level $24$ and weight $2$. Funct. Approx. Comment. Math. 57 (2017), no. 2, 205--234. doi:10.7169/facm/1628.

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