The average behavior of Fourier coefficients of cusp forms over sparse sequences

Let λ ( n ) λ (n) be the n n th normalized Fourier coefficient of a holomorphic Hecke eigenform f ( z ) ∈ S k ( Γ ) f(z)∈ Sₖ(Γ ) . In this paper we are interested in the average behavior of λ 2 ( n ) λ ^2(n) over sparse sequences. By using the properties of symmetric power L L -functions and their Rankin-Selberg L L -functions, we are able to establish that for any ε > 0 ε >0 , \[ ∑ n ≤ x λ 2 ( n j ) = c j − 1 x + O ( x 1 − 2 ( j + 1 ) 2 + 2 + ε