On the coefficients of automorphic representations over polynomials

Let \(\pi \) $\pi$ be a cuspidal automorphic representation of \(\textrm{GL}_2(\mathbb {A}_\mathbb {Q})\) ${\text{GL}}_2({\mathbb{A}}_{\mathbb{Q}})$ associated to holomorphic forms with Fourier coefficients \(a_{ \pi }(n)\) $a_{\pi }(n)$ . Consider an automorphic representation \(\Pi \) $\mathrm{\Pi }$ which is equivalent to \(\textrm{sym}^m \pi \) ${\text{sym}}^m\pi$ or \(\pi \times \textrm{sym}^m \pi \) $\pi \times {\text{sym}}^m\pi$ . We establish uniform upper bounds for \(\sum _{n\leqslant X} |a_{\Pi } (|f(n)|)|\) $\sum _{n⩽X}^{}|a_{\mathrm{\Pi }}(|f(n)|)|$ , where \(f(x)\in \mathbb {Z}[x]\) $f(x)\in \mathbb{Z}[x]$ is a polynomial of arbitrary degree. This builds on the work of Chiriac and Yang, and refines one of their results.