Let \(\lambda _f(n)\) be the Fourier coefficients of a normalized Hecke eigenform f of weight k for the full modular group \(SL(2, \mathbb {Z})\) . For \(j \ge 2\) , let \(\lambda _{sym^jf}(n)\) be the coefficients of the Dirichlet series of the jth symmetric power L-function associated with f. Let \(\sigma (n)\) and \(\phi (n)\) be the sum of the divisors function and the Euler totient function, respectively. For given real numbers b and c, we prove asymptotic results for the higher power moments of \(\lambda _f^j(n)\sigma ^b(n)\phi ^c(n)\) and for the second power moments of \(\lambda ^2_{sym^jf}(n)\sigma ^b(n)\phi ^c(n)\) over sequences of positive integers given by two distinct polynomials.
