On the distribution of the Fourier coefficients over two sparse sequences

Let $j\geq 3$ be any fixed integer and $f$ be a primitive holomorphic cusp form of even integral weight $\kappa\geq 2$ for the full modular group $SL(2,\mathbb{Z})$. We write $\lambda_{{\rm{sym}^j }f}(n)$ for the $n^\text{th}$ normalized Fourier coefficient of $L(s,{\rm{sym}}^j f)$. In this article, we establish asymptotic formulae for the discrete sums of the Fourier coefficients $\lambda_{{\rm{sym}}^j f}^2(n)$ over two sparse sequence of integers, which can be written as the sum of four integral squares and the sum of six integral squares, with refined error terms.