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Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms. (English) Zbl 1364.11095

Some new results on non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms are obtained.
Authors’ abstract: In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form \(\mathfrak{f}\) with half-integral weight:
\([-
\)] The first negative coefficient of the sequence \(\{\mathfrak{a}_{\mathfrak{f}}(t n^2) \}_{n \in \mathbb{N}}\),
\([-
\)] The number of coefficients \(\mathfrak{a}_{\mathfrak{f}}(t n^2)\) of same signs,
\([-
\)] Non-vanishing of coefficients \(\mathfrak{a}_{\mathfrak{f}}(t n^2)\) in short intervals and in arithmetic progressions, where \(\mathfrak{a}_{\mathfrak{f}}(n)\) is the \(n\)th Fourier coefficient of \(\mathfrak{f}\) and \(t\) is a square-free integer such that \(\mathfrak{a}_{\mathfrak{f}}(t) \neq 0\).

MSC:

11F25 Hecke-Petersson operators, differential operators (one variable)
11F37 Forms of half-integer weight; nonholomorphic modular forms
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