Murmurations of Hecke Eigenvalues at Prime Squares in the Weight Aspect

We study the root-number-weighted average of the normalized Hecke eigenvaluesλf(p2) over level-one eigenforms f, with smooth weights in the weight k (window width Kθaround K) and in p2/N, where N = ((K −1)/4π)2. Assuming GRH for Dirichlet L-functionsand 7/8 < θ < 1, we show that this average converges, as K → ∞, to an explicit limit, with afinite-K main term that approaches the limit at rate K−2(1−θ). Unconditionally, we identify thelimit with an explicit ordered sum of point masses at y = (c/m)2 whose weights are averagesof the Kloosterman sums S(a2,1;c). The limit differs from the corresponding limit for primes,and the difference is arithmetic. The unconditional part of the argument (the local values ofthe densities, the Kloosterman evaluation, the global identity and the identification of the limit)has been formally verified in Lean 4 with Mathlib