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.

We investigate the triple correlation sums of coefficients of $\theta$-series and prove an asymptotic formula with power-saving error term. As a result, we present that this type of sum is non-trivial in the regime $H\ge X^{2/3+\epsilon }$.

Let $p$ be a prime. In this paper, we study the sum

$$\sum _{m\ge 1}\sum _{n\ge 1}{a}_n{\lambda }_g(m){\lambda }_f(m+pn)U\left(\frac{m}{X}\right)V\left(\frac{n}{H}\right)$$

for any newforms $g\in {\mathcal{B}}_k(1)$ (or ${\mathcal{B}}_{\lambda }^{\ast }(1)$) and $f\in {\mathcal{B}}_k(p)$ (or ${\mathcal{B}}_{\lambda }^{\ast }(p)$), with the aim of determining the explicit dependence on the level, where $\mathbf{a}=\{a_n\in \mathbb{C}\}$ is an arbitrary complex sequence. As a result, we prove a uniform bound with respect to the level parameter $p$, and present that this type of sum is non-trivial for any given $H,X\ge 2$.