
On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms
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.\newline
(i) For any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n\leq x}\lambda_f^5(n) \ll_{f,\varepsilon}x^{\frac{15}{16}+\varepsilon}
\quad\text{and}\quad\sum_{n\leq x}\lambda_f^7(n) \ll_{f,\varepsilon}x^{\frac{63}{64}+\varepsilon}.
\end{equation*}
(ii) If $\operatorname{sym}^3\pi_f \ncong \operatorname{sym}^3\pi_g$, then for any $\varepsilon>0$, we have
\begin{equation*}
\sum_{n \leq x}\lambda_f^3(n)\lambda_g^3(n)\ll_{f,\varepsilon}x^{\frac{31}{32}+\varepsilon};
\end{equation*}
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^2(n)=cx\log x+c'x+O_{f,\varepsilon}\bigl(x^{\frac{31}{32}+\varepsilon}\bigr);
\]
If $\operatorname{sym}^2\pi_f \ncong \operatorname{sym}^2\pi_g$ and $\operatorname{sym}^4\pi_f \ncong \operatorname{sym}^4\pi_g$, then for any $\varepsilon>0$, we have
\[
\sum_{n \leq x}\lambda_f^4(n)\lambda_g^4(n)=xP(\log x)+O_{f,\varepsilon}\bigl(x^{\frac{127}{128}+\varepsilon}\bigr),
\]
where $P(x)$ is a polynomial of degree $3$.
Keywords: Fourier coefficients of cusp forms, symmetric power $L$function Categories:11F30, , , , 11F11, 11F66 