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April 1, 2026American Journal of Mathematics0 citations

On the first sign change of Fourier coefficients of cusp forms

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GCGuohua ChenYLYuk-Kam LauYWYingnan Wang

Key Points

  • The research aims to identify the location of the first sign change of Fourier coefficients in holomorphic cusp forms with real coefficients.
  • Introduced a new method based on the Eisenstein series.
  • Avoided reliance on Atkin-Lehner theory for cuspidal newforms.
  • Applicable to integral weight and half-integral weight cusp forms.
  • Focused on the congruence subgroup Gamma_0(N) for any level N.
  • Provided new insights into the sign change locations of Fourier coefficients.
  • Clarified applications for a broader range of cusp forms than previously understood.

Abstract

abstract: We present a variant of the current widely-used method initiated by Choie and Kohnen in the study of the location of the first sign change of the Fourier coefficients of a holomorphic cusp form when all the coefficients are real. This version circumvents the use of Atkin-Lehner theory of cuspidal newforms, instead utilizing the Eisenstein series, and it applies directly to cases including integral weight cusp forms on the congruence subgroup ₀ (N) of any level N as well as half-integral weight cusp forms.

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Cite This Study

Chen et al. (2026) studied this question.

synapsesocial.com/papers/69cd7a1b5652765b073a6f11https://doi.org/10.1353/ajm.2026.a986597
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Also Consider

Synapse has enriched 5 closely related papers on similar clinical questions. Consider them for comparative context:

  1. 1Fourier coefficients of cusp forms on special sequences2024
  2. 2Fourier coefficients and cuspidality of modular forms: a new approach2024
  3. 3On Signs of Fourier Coefficients on GL( n )2025
  4. 4On the non-vanishing of Fourier coefficients of half-integral weight cuspforms2024
  5. 5A Fourier-Jacobi Dirichlet series for cusp forms on orthogonal groups2024