Let {{S}_{k}}(\Gamma )${{S}_{k}}(\Gamma )$ be the space of holomorphic cusp forms of even integral weight k$k$ for the full modular group. Let {{\lambda }_{f}}(n)${{\lambda }_{f}}(n)$ and {{\lambda }_{g}}(n)${{\lambda }_{g}}(n)$ be the n$n$-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms f(z),\,g(z)\,\in \,{{S}_{k}}(\Gamma )$f(z),\,g(z)\,\in \,{{S}_{k}}(\Gamma )$, respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.

(i)For any \varepsilon \,>\,0$\varepsilon \,>\,0$, we have

\sum\limits_{n\le x}{\lambda _{f}^{5}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{15}{16}+\varepsilon }}\text{and}\sum\limits_{n\le x}{\lambda _{f}^{7}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{63}{64}+\varepsilon }}. $$\sum\limits_{n\le x}{\lambda _{f}^{5}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{15}{16}+\varepsilon }}\text{and}\sum\limits_{n\le x}{\lambda _{f}^{7}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{63}{64}+\varepsilon }}.$$

(ii)If \text{sy}{{\text{m}}^{3\,}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{3\,}}{{\pi }_{g}}\,$\text{sy}{{\text{m}}^{3\,}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{3\,}}{{\pi }_{g}}\,$, then for any \varepsilon \,>\,0$\varepsilon \,>\,0$, we have

\sum\limits_{n\le x}{\lambda _{f}^{3}(n)\lambda _{g}^{3}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{31}{32}+\varepsilon }}; $$\sum\limits_{n\le x}{\lambda _{f}^{3}(n)\lambda _{g}^{3}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{31}{32}+\varepsilon }};$$

If \text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$, then for any \varepsilon \,>\,0$\varepsilon \,>\,0$, we have

\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{2}(n)}=cx\log x+{c}'x+{{O}_{f,\varepsilon }}({{x}^{\frac{31}{32}+\varepsilon }}); $$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{2}(n)}=cx\log x+{c}'x+{{O}_{f,\varepsilon }}({{x}^{\frac{31}{32}+\varepsilon }});$$

If \text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$ and \text{sy}{{\text{m}}^{4}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{4}}{{\pi }_{g}}$\text{sy}{{\text{m}}^{4}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{4}}{{\pi }_{g}}$, then for any \varepsilon \,>\,0$\varepsilon \,>\,0$, we have

\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{4}(n)}=xP(\log x)+{{O}_{f,\varepsilon }}({{x}^{\frac{127}{128}+\varepsilon }}), $$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{4}(n)}=xP(\log x)+{{O}_{f,\varepsilon }}({{x}^{\frac{127}{128}+\varepsilon }}),$$

where P\left( x \right)$P\left( x \right)$ is a polynomial of degree 3.