Joint level-weight murmurations for holomorphic newforms

Abstract

We establish an unconditional joint level-weight murmuration law for holomorphic newforms of squarefree level. The level N is averaged at size X, the even weight k at size K, and primes are measured in the coordinate p divided by N times ((k-1)/(4 pi)) squared. After normalization, the root-number weighted prime trace converges to a locally finite measure supported on rational squares, with explicitly given Euler-product masses. The convergence is uniform for X^epsilon <= K <= X^(1-epsilon), and therefore holds for K = X^rho for every fixed 0 < rho < 1. Starting from the exact trace formula, we first sum over the weight, truncate the resulting quadratic L-values using Burgess&#39;s bound, and average the remaining periodic functions over squarefree levels. Neumann summation converts the resulting Bessel series into the atomic measure. No hypothesis on zeros of Dirichlet or automorphic L-functions is used.