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The purpose of this paper is to obtain asymptotics of shifted sums of Heckeeigenvalue squares on average. We show that for $X^{\frac{2}{3}+\epsilon} < H<X^{1-\epsilon},$ there are constants $B_{h}$ such that $$ \sum_{X\leq n \leq2X} \lambda_{f}(n)^{2}\lambda_{f}(n+h)^{2}-B_{h}X=O_{f,A,\epsilon}\big(X (\logX)^{-A}\big)$$ for all but $O_{f,A,\epsilon}\big(H(\log X)^{-3A}\big)$ integers$h \in [1,H]$ where $\{\lambda_{f}(n)\}_{n\geq1}$ are normalized Heckeeigenvalues of a fixed holomorphic cusp form $f.$ Our method is based on theHardy-Littlewood circle method. We divide the minor arcs into two parts $m_{1}$and $m_{2}.$ In order to treat $m_{2},$ we use the Hecke relations, a bound ofMiller to apply some arguments from a paper of Matom\"{a}ki, Radziwill and Tao.We apply Parseval's identity and Gallagher's lemma so as to treat $m_{1}.$

On the asymptotics of the shifted sums of Hecke eigenvalue squares