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

Abstract

Let $f$ be a normalized Hecke eigen cusp form of even integral weight $k$ for the full modular group $SL(2,Z)$. For integers $j\ge 2$, let ${\lambda }_{{\mathrm{Sym}}^{j}f}\left(m\right)$ 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 ${\lambda }_{{\mathrm{Sym}}^{j}f}\left(m\right)$ for the indices $m$ that are the sum of two squares in the interval $[1,x]$ for sufficiently large $x$.