Sign changes in Fourier coefficients of the symmetric power $L$-functions on sums of two squares

TL;DR

This paper investigates the sign changes of Fourier coefficients of symmetric power L-functions associated with modular forms, specifically for indices that are sums of two squares, providing quantitative results for large intervals.

Contribution

It offers a new quantitative analysis of sign changes of Fourier coefficients of symmetric power L-functions at sums of two squares, extending understanding of their oscillatory behavior.

Findings

Quantitative bounds on the number of sign changes for large x

Sign change results specifically for indices that are sums of two squares

Enhanced understanding of Fourier coefficient oscillations in symmetric power L-functions

Abstract

Let $f$ be a normalized primitive Hecke eigen cusp form of even integral weight $k$ for the full modular group $SL(2,\mathbb{Z})$. For integers $j \geq 2$, let $\lambda_{sym^j f}(m)$ denote the $m$th Fourier coefficient of the $j$th symmetric power $L$-function associated with $f$. We give a quantitative result on the number of sign changes of $\lambda_{sym^j f}(m)$ for the indices $m$ that are the sum of two squares in the interval $[1,x]$ for sufficiently large $x$.