Level aspect exponential sums involving Fourier coefficients of symmetric-square lifts

Fix an integer \(\kappa\ge 2\) $\kappa \ge 2$ . Let \(P\ge 2\) $P\ge 2$ be a prime, and \(F\) $F$ be thesymmetric-square lift of a Hecke newform \(f\in \mathcal{S}^ {\ast} _\kappa(P)\) $f\in {\mathcal{S}}_{\kappa }^{*}(P)$ . We study the exponential sum \(\begin{aligned}\mathscr{L}_F(\alpha)=\sum_{n\sim N} A_F(n,1)e(n \alpha) \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{L}}_F(\alpha )=\sum _{n\sim N}{A}_F(n,1)e(n\alpha )\end{array}$$ by implementing an average over a family in such a way to investigate the bestpossible magnitude of the level aspect bound for \(\mathscr{L}_F(\alpha)\) ${\mathcal{L}}_F(\alpha )$ . We prove a uniformbound with respect to any \(\alpha \in \mathbb{R}\) $\alpha \in \mathbb{R}$ and the level parameter \(P\) $P$ , and present thatthere exist certain forms with fairly strong oscillations in \(\mathscr{L}_F(\alpha)\) ${\mathcal{L}}_F(\alpha )$ , if the associatedlevel of \(f\) $f$ is allowed to vary. As applications, we consider the shifted convolutionsums for \( \mathrm{GL} (3)\times \mathrm{GL} (d)\) $\mathrm{GL}(3)\times \mathrm{GL}(d)$ , for any \(d\ge 2\) $d\ge 2$ , in a family as well as theWaring-Goldbachproblem associated to Fourier coefficients of \( \mathrm{SL} (3,\mathbb{Z})\) $\mathrm{SL}(3,\mathbb{Z})$ -Maa \(\beta\) $\beta$ forms.