On triple correlation sums of Fourier coefficients of cusp forms

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 $H,X\ge 2$.