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

## TL;DR

This paper investigates the sign distribution of Fourier coefficients of newforms, especially at prime powers, and determines the convergence properties of related Dirichlet series involving symmetric power $L$-functions.

## Contribution

It provides new insights into the sign patterns of Fourier coefficients at prime powers and identifies the abscissas of absolute convergence for associated Dirichlet series.

## Key findings

- Distribution of signs of Fourier coefficients at prime powers analyzed
- Abscissas of absolute convergence for related Dirichlet series determined
- Results contribute to understanding the behavior of symmetric power $L$-functions

## Abstract

We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for $\Gamma_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the distribution of the signs of $\{a_f(p^m)\}_p$, where $p$ 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 $L$-functions.