On the exponential sums estimates related to Fourier coefficients of $G{L}_3$ Hecke-Maaß forms

Let $F$ be a normlized Hecke-Maaß form for the congruent subgroup ${\mathrm{\Gamma }}_0(N)$ with trivial nebentypus. In this paper, we study the problem of the level aspect estimates for the exponential sum

$${\mathcal{L}}_F(\alpha )=\sum _{n\le X}{A}_F(n,1)e(n\alpha ).$$

As a result, we present an explicit non-trivial bound for the sum ${\mathcal{L}}_F(\alpha )$ in the case of $N=P$. In addition, we investigate the magnitude for the non-linear exponential sums with the level being explicitly determined from the sup-norm's point of view as well.