Power Sums of Symmetric Square Coefficients over Diverse Arithmetic Sequences

We investigate the average behaviour of the Fourier coefficients of symmetric square L-functions, λ sym² f(n), attached to holomorphic Hecke eigenforms. Recent work by Tiwari and Godara (P. Tiwari and N. K. Godara, On the power moments of hybrid arithmetic functions associated with the Hecke eigenvalues, Ramanujan J. 66 (2025), 46) has studied related sums over sequences generated by polynomials g(x) and h(x). We extend this line of inquiry by analysing the m-th power sums over these same two polynomial sequences, ∑ _n=g(a)+1 ≤ x λ sym² f(nᵐ) and ∑ _n=h(a)+1 ≤ x λ sym² f(nᵐ), alongside the sum ∑ _n=Q(x) ≤ x λ sym² f(nᵐ) over integers n represented by primitive positive-definite binary quadratic forms. In all three cases, the underlying multiplicative representation function $r(n)$ is related to coefficients of Eisenstein series. We establish unified asymptotic formulas for all m ≥ 2 for all three of these sequences.