ON THE GENERAL TRIPLE CORRELATION SUMS FOR GL2×GL2×GL2

Fix X≥2. Let f be a Hecke newform of prime level p. We investigate the general triple correlation sum ∑h≥1∑l≥1∑n≥1λf(n)λf(n+h)λf(n+l)U(nX)V(hH)R(lL) for H,L≥1 in the level aspect. As a result, we prove a nontrivial bound for any H,L satisfying that L>X1∕4 and max {L3X−2,L,X1∕4}<H<min {X2∕3L1∕3,L2}. It can be shown that there exist certain newforms such the nontrivial bound for the triple sum can be achieved, so long as max {H,L}≥X1∕4+𝜀. Particularly, whenever L=H, we present a nontrivial estimate for any p such that H2∕X≤p<min {H2X−1∕2,H}, and further obtain the more significant cancellations for these sums in the different segments of H.