Let $f$ denote a normalized primitive holomorphic cusp form of even integral weight $l$ for congruence group ${\mathrm{\Gamma }}_0(N)$ and ${\lambda }_f(n)$ the $n$-th Hecke eigenvalue of $f$. We consider the higher-power moments of $S_f(x,N)≔{\sum }_{n\le x}{\lambda }_f(n)$ and derive asymptotic formulas for the third through seventh power moments, improving the previous results.