On the second moment of Fourier coefficients of symmetric power $L$-function

Let $f(z)$ be the normalized primitive holomorphic Hecke eigenforms of even integral weight $k$ for the full modular group SL$(2,\mathbb{Z})$. In this paper, we investigate the second moment of Fourier coefficients of symmetric power $L$-function $L(s,{\mathrm{\text{sym}}}^{j}f)$ for $j\ge 2$ and improve on previous error estimates. We also consider the second moment of ${\lambda }_f(n^2)$.