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On the coefficients of symmetric power $L$ $L$ -functions

School of Mathematical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar, P.O. Jatni, Khurda 752 050, Odisha, India

E-mail Address: jaban@niser.ac.in

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Karam Deo Shankhadhar

Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462 066, Madhya Pradesh, India

E-mail Address: karamdeo@iiserb.ac.in

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, and

G. K. Viswanadham

Department of Mathematics, Indian Institute of Technology Bombay, Mumbai-400 076, India

E-mail Address: vissu35@gmail.com

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https://doi.org/10.1142/S1793042118500495Cited by:9 (Source: Crossref)

Abstract

We study the signs of the Fourier coefficients of a newform. Let $f$ $f$ be a normalized newform of weight $k$ $k$ for $Γ_{0} (N)$ $Γ_{0} (N)$ . Let $a_{f} (n)$ $a_{f} (n)$ be the $n$ $n$ th Fourier coefficient of $f$ $f$ . For any fixed positive integer $m$ $m$ , we study the distribution of the signs of ${a_{f} (p^{m})}_{p}$ ${a_{f} (p^{m})}_{p}$ , where $p$ $p$ runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power $L$ $L$ -functions.

Keywords:

AMSC: 11F11, 11F30, 11M41

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