On the distribution of the Fourier coefficients over two sparse sequences

Let $j\ge 3$ be any fixed integer and $f$ be a primitive holomorphic cusp form of even integral weight $\kappa \ge 2$ for the full modular group $SL(2,\mathbb{Z})$. We write ${\lambda }_{{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f}(n)$ for the $n^{\text{th}}$ normalized Fourier coefficient of $L(s,{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f)$. In this article, we establish asymptotic formulae for the discrete sums of the Fourier coefficients ${\lambda }_{{\mathrm{s}\mathrm{y}\mathrm{m}}^{j}f}^2(n)$ 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.