Let
${{S}_{k}}(\Gamma )$
be the space of holomorphic cusp forms of even integral weight
$k$
for the full modular group. Let
${{\lambda }_{f}}(n)$
and
${{\lambda }_{g}}(n)$
be the
$n$
-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms
$f(z),\,g(z)\,\in \,{{S}_{k}}(\Gamma )$
, 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
$\varepsilon \,>\,0$
, we have
$$\sum\limits_{n\le x}{\lambda _{f}^{5}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{15}{16}+\varepsilon }}\text{and}\sum\limits_{n\le x}{\lambda _{f}^{7}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{63}{64}+\varepsilon }}.$$
(ii)If
$\text{sy}{{\text{m}}^{3\,}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{3\,}}{{\pi }_{g}}\,$
, then for any
$\varepsilon \,>\,0$
, we have
$$\sum\limits_{n\le x}{\lambda _{f}^{3}(n)\lambda _{g}^{3}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{31}{32}+\varepsilon }};$$
If
$\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$
, then for any
$\varepsilon \,>\,0$
, we have
$$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{2}(n)}=cx\log x+{c}'x+{{O}_{f,\varepsilon }}({{x}^{\frac{31}{32}+\varepsilon }});$$
If
$\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$
and
$\text{sy}{{\text{m}}^{4}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{4}}{{\pi }_{g}}$
, then for any
$\varepsilon \,>\,0$
, we have
$$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{4}(n)}=xP(\log x)+{{O}_{f,\varepsilon }}({{x}^{\frac{127}{128}+\varepsilon }}),$$
where
$P\left( x \right)$
is a polynomial of degree 3.