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Modular forms which behave like theta series. (English) Zbl 1036.11503

Summary: We determine all modular forms of weights \(36\leq k\leq 56\), \(4\mid k\), for the full modular group \(\text{SL}_2(\mathbb Z)\) which behave like theta series, i.e., which have in their Fourier expansions the constant term \(1\) and all other Fourier coefficients are non-negative rational integers. In fact, we give convex regions in \({\mathbb R}^3\) (resp. in \({\mathbb R}^4\)) for the cases \(k = 36, 40 \hbox{~and~} 44\) (resp. for the cases \(k = 48, 52 \hbox{~and~} 56\)). Corresponding to each lattice point in these regions, we get a modular form with the above property. As an application, we determine the possible exceptions of quadratic forms in the respective dimensions.

MSC:

11F11 Holomorphic modular forms of integral weight
11Y35 Analytic computations
11F27 Theta series; Weil representation; theta correspondences
11F30 Fourier coefficients of automorphic forms
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References:

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