Murmurations of Hecke eigenvalues at prime cubes in the weight aspect

Murmurations of Hecke eigenvalues at prime cubes in the weight aspectMatthew Gifford, November 2026 (preprint) Abstract We study the root-number-weighted average of the normalized Hecke eigenvalues λf(p³) over level-one eigenforms f, with smooth weights in the weight k (window width Kθ around K) and in p³/N, where N = ((K−1)/4π)². Assuming GRH for Dirichlet L-functions and 8/9 < θ < 1, we show that this average converges, as K → ∞, to an explicit limit. A finite-K main term approaches the limit at rate K−2(1−θ). Under the announced, not yet refereed, zero-free half-planes Re s > 7/8 and Re s > 11/12, the same holds for 71/72 < θ < 1 and 143/144 < θ < 1. Unconditionally, we identify the limit with a sum of point masses at y = (c/m)², absolutely convergent over c, whose weights are averages of the Kloosterman sums S(m²b³, 1; c). Unlike the cases of primes and prime squares, these weights depend on m mod c and not only on (m, c). This comes from a local identity, valid for every exponent j and every prime. The local densities of the discriminants t² − 4bj have a Fourier expansion whose coefficients are Kloosterman averages twisted by u², and these coefficients are multiplicative. For j = 3: the density at a prime ℓ ∤ 6t is a point count on the CM curve Y² = X³ + 16t²; the twist is genuine at ℓ ≡ 1 (mod 3); the Euler product at t = 0 is ζ(2). For j = 2, the same identity reproves the Kloosterman evaluation behind the prime-square limit without case analysis. Main results Theorem A (GRH): convergence of the statistic M₃(K) to Λ₃ = (2ζ(2))⁻¹ Σt f₃(t) J(t)/J(0) for 8/9 < θ < 1, with the explicit finite-K main term. Theorem B (unconditional): Λ₃ as a sum of point masses with twisted Kloosterman weights. Theorem C: twisted Kloosterman duality for the local densities, for every exponent j and every prime. Theorem D: the global identity f₃(t) = Πℓ Eℓ(t) = Σc W(c, t), with a uniform bound and f₃(0) = ζ(2). Proposition 11: the ranges of θ under zero-free half-planes. Verification Numerical checks. Every identity was tested with code written separately from the proofs. The arguments were audited by separate sessions, including one blind audit of the whole written proof. Lean. Everything except Lemma 6 (the trace formula for Tp³) has been formalized in Lean 4 with Mathlib, Theorem A included. The trace formula, the class-number and dimension formulas, and the two GRH inputs enter as explicit hypotheses; no axioms are added beyond Lean's standard three. Archive. Code, logs, audit reports and the Lean files are at doi:10.5281/zenodo.24148150. Use of AI tools This work was carried out in collaboration with an AI assistant (Claude, Anthropic), which drafted the proofs, the numerical code, the Lean formalization and the manuscript under the author's direction. Nothing has yet been refereed by a specialist; the author takes responsibility for the content.