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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Sign changes of coefficients of half-integral weight Hecke eigenforms
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by Chenran Xu
Proc. Amer. Math. Soc. 151 (2023), 4633-4642
DOI: https://doi.org/10.1090/proc/16527
Published electronically: August 4, 2023

Abstract:

Let $\mathfrak {f}$ be a cusp form of half-weight $k+1/2$ and at most quadratic nebentype character whose Fourier coefficients are denoted as $\mathfrak {a}_\mathfrak {f}(n)$. We study sign changes of the family $\{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}}$ where $t$ is a square-free number and $p$ runs through the prime numbers. By taking use of the relationship between the cusp form $\mathfrak {f}$ of half-integral weight and the Shimura lift $f_t$, we establish that under certain conditions, the Hecke eigenforms of half-integral weight could be determined by sign changes of the sequence $\{\mathfrak {a}_\mathfrak {f}(tp^2)\}_{p\in \mathbb {P}}$.
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Bibliographic Information
  • Chenran Xu
  • Affiliation: School of Mathematics, Shandong University, Jinan, Shandong 250100, People’s Republic of China
  • Received by editor(s): June 10, 2022
  • Received by editor(s) in revised form: October 26, 2022, January 20, 2023, and March 23, 2023
  • Published electronically: August 4, 2023
  • Communicated by: Amanda Folsom
  • © Copyright 2023 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 151 (2023), 4633-4642
  • MSC (2020): Primary 11F30, 11F41, 11F03, 11E45, 11-04
  • DOI: https://doi.org/10.1090/proc/16527
  • MathSciNet review: 4634869