The distribution of Fourier coefficients of symmetric square L-functions over arithmetic progressions

Let \(L(s, \mathrm{sym^2}f)\) $L(s,{\mathrm{sym}}^{2}f)$ be the corresponding symmetric square L-function associated to f(z), where f(z) is a primitive holomorphic cusp form of even integral weight k for the full modular group. Suppose that \(\lambda _{\mathrm{sym^2}f} (n)\) ${\lambda }_{{\mathrm{sym}}^{2}f}(n)$ is the nth normalized Fourier coefficient of \(L(s, {\mathrm{sym^2}f})\) $L(s,{\mathrm{sym}}^{2}f)$ . In this paper, we use the function equation and the large sieve inequality to study the asymptotic behaviour of the sums \(\begin{aligned} \sum _{\begin{array}{c} n\leqslant x \\ n\equiv a(\textrm{mod}\ q) \end{array}}\lambda ^{j}_{\mathrm{sym^2}f}(n), 2\leqslant j\leqslant 4. \end{aligned}\) $$\begin{array}{c}\hfill \sum _{\begin{array}{c}n⩽x\\ n\equiv a(\text{mod}q)\end{array}}{\lambda }_{{\mathrm{sym}}^{2}f}^j(n),2⩽j⩽4.\end{array}$$