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Average estimates and sign change of Fourier coefficients of cusp forms at integers represented by binary quadratic form of fixed discriminant

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Abstract

In this article, we establish an average behaviour of the Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant \(D < 0\) when the class number \(h(D) = 1\). We also obtain a quantitative result for the number of sign changes of the sequence of Fourier coefficients a(n) of the Hecke eigenforms, where n is represented by any primitive integral positive definite binary quadratic form of fixed discriminant \(D < 0\) when the class number \(h(D) = 1\) in the interval (x, 2x], for sufficiently large x.

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Acknowledgements

This work is a part of the author’s thesis [30, Chapter 4]. The author would like to thank Prof. B Ramakrishnan for encouraging us to work on this problem and also for his guidance and suggestions during the preparation of the manuscript. We thank HRI, Prayagraj for providing financial support through an Infosys grant.

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  1. Lalit Vaishya

    Present address: Present address: The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai, 600 113, India

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  1. Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Prayagraj, 211 019, India

    Lalit Vaishya

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  1. Lalit Vaishya
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Correspondence to Lalit Vaishya.

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Communicated by Sanoli Gun.

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Vaishya, L. Average estimates and sign change of Fourier coefficients of cusp forms at integers represented by binary quadratic form of fixed discriminant. Proc Math Sci 134, 17 (2024). https://doi.org/10.1007/s12044-024-00782-6

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