Sign of Fourier coefficients of half-integral weight modular forms in arithmetic progressions

Abstract

Let f be a half-integral weight cusp form of level 4N for odd and squarefree N and let a(n) denote its nth normalized Fourier coefficient. Assuming that all the coefficients a(n) are real, we study the sign of a(n) when n runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers \(n\leqslant x\) such that \(a(n)>n^{-\alpha }\) where x and \(\alpha \) are positive and f is not necessarily a Hecke eigenform.

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Notes

  1. 1.

    Here and in the rest of the paper, a positive proportion means a number of \(a\,[p]\) which is \(\gg p\).

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Acknowledgements

The author would like to express his gratitude to Florent Jouve and Guillaume Ricotta for their many helpful comments and useful suggestions.

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Correspondence to Corentin Darreye.

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Darreye, C. Sign of Fourier coefficients of half-integral weight modular forms in arithmetic progressions. Res. number theory 6, 46 (2020). https://doi.org/10.1007/s40993-020-00225-x

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Keywords

  • Modular form
  • Fourier coefficient
  • Arithmetic progression
  • Sign