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Let $\eta>0$ be a fixed positive number, let $N$ be a sufficiently largenumber. In this paper, we study the second moment of the sum of Heckeeigenvalues over primes in short intervals (whose length is $\eta \log N$) onaverage (with some weights) over the family of weight $k$ holomorphic Heckecusp forms. We also generalize the above result to Hecke-Maass cusp forms for$SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewoodprime 2-tuples conjecture, we calculate the exact values of the mean values.

On sum of Hecke eigenvalue squares over primes in very short intervals