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

Fix an integer \(\kappa\ge 2\). Let \(P\ge 2\) be a prime, and \(F\) be the symmetric-square lift of a Hecke newform \(f\in \mathcal{S}^ {\ast} _\kappa(P)\). We study the exponential sum

$$\begin{aligned}\mathscr{L}_F(\alpha)=\sum_{n\sim N} A_F(n,1)e(n \alpha) \end{aligned}$$

by implementing an average over a family in such a way to investigate the best possible magnitude of the level aspect bound for \(\mathscr{L}_F(\alpha)\). We prove a uniform bound with respect to any \(\alpha \in \mathbb{R}\) and the level parameter \(P\), and present that there exist certain forms with fairly strong oscillations in \(\mathscr{L}_F(\alpha)\), if the associated level of \(f\) is allowed to vary. As applications, we consider the shifted convolution sums for \( \mathrm{GL} (3)\times \mathrm{GL} (d)\), for any \(d\ge 2\), in a family as well as theWaring-Goldbach problem associated to Fourier coefficients of \( \mathrm{SL} (3,\mathbb{Z})\)-Maa\(\beta\) forms.