Murmurations of modular forms in the weight aspect

Abstract

We prove the existence of "murmurations" in the family of holomorphic modular forms of level $1$ and weight $k\to \infty$, that is, correlations between their root numbers and Hecke eigenvalues at primes growing in proportion to the analytic conductor. This is the first demonstration of murmurations in an archimedean family.

30 pages, to appear in Algebra & Number Theory