`Fix $X\ge 2$. Let $f$ be a `

`Hecke`

` `

`newform`

` of prime level $p$. In this paper, we investigate the general triple correlation sum \[ \sum_{h\ge 1}\sum_{l\ge 1}`

`\sum_{n\ge 1} \lambda_{f}(n) \lambda_{f}(n+h) \lambda_{f}(n+l)\,U{\lf(\f{n}{X}\ri) }V{\lf (\f{h}{H}\ri)}R{\lf (\f{l}{L}\ri)} \] for $H,L\ge 1$ in the level aspect. As a result, we prove a non-trivial bound for any $H,L$ satisfying that $L> X^{1/4}$ and $\max\{L^3X^{-2},\sqrt{L},X^{1/4}\}< H< \min\{ X^{2/3}L^{1/3},L^2\}$. It can be shown that, for $\max\{H,L\}\ge X^{1/4+\varepsilon}$, the non-trivial bounds for the triple sum are achieved up to certain newforms. Particularly, whenever $L=H$, we present a non-trivial estimate for any $p $ such that $H^2/X\le p< \min\{H^2X^{-1/2}, H\}$, and obtain sharp quantitative bounds in different segments of $H$.`

**This work is now UNDER REVIEW on the Journal: ''The Rocky Mountain Journal of Mathematics", which was submitted on 11 Jul 2022. @All rights reserved.**