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Quadratic minima and modular forms. (English) Zbl 0916.11025
The main results are upper bounds on the size of the gap between the constant term and the next nonzero Fourier coefficient of an entire modular form of given weight for \(\Gamma_0(2)\), generalizing C. L. Siegel’s result [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1970, 15-56 (1970; Zbl 0225.10031)] for the full modular group. Numerical evidence indicates that a sharper bound holds for the weights \(n\equiv 2\text{ mod }4\). As an application, upper bounds are derived for the minimum positive integer represented by level-two even positive definite quadratic forms.

MSC:
11F11 Holomorphic modular forms of integral weight
11E20 General ternary and quaternary quadratic forms; forms of more than two variables
11F30 Fourier coefficients of automorphic forms
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