On the first sign change of Fourier coefficients of cusp forms. (English) Zbl 1441.11093
Summary: Let \(f\) be a nonzero cusp form of even integral weight \(k \geqslant 2\) on the Hecke congruence subgroup \(\Gamma_0(N)\) with \(N\) squarefree. Suppose that the normalized Fourier coefficients \(\lambda_f(n)\) of \(f\) are real. We prove that the first sign change of \(\lambda_f(n)\) occurs in the range \(n \ll(k N)^{2 + \varepsilon}\). This improves upon the earlier result of Y. Choie and W. Kohnen [Am. J. Math. 131, No. 2, 517–543 (2009; Zbl 1254.11044)].
Citations:
Zbl 1254.11044References:
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| [2] | Choie, Y.; Kohnen, W., The first sign change of Fourier coefficients of cusp forms, Amer. J. Math., 131, 2, 517-543 (2009) · Zbl 1254.11044 |
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