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Let $lambda (n)$ be the $n$th normalized Fourier coefficient of a holomorphic Hecke eigenform $f(z)in S_{k}(Gamma )$. In this paper we are interested in the average behavior of $lambda ^2(n)$ over sparse sequences. By using the properties of symmetric power $L$-functions and their Rankin-Selberg $L$-functions, we are able to establish that for any $varepsilon >0$, $$sum _{n leq x}lambda ^2(n^j)=c_{j-1} x+Oleft (x^{1-frac {2}{(j+1)^2+2}+varepsilon }right ),$$ where $j=2,3,4.$ [ABSTRACT FROM AUTHOR]