Universal prime and prime-power murmuration across self-dual GL(2) families

Establish that, for every specified self-dual family of primitive GL(2) forms ordered by analytic conductor, the root-number-weighted averages of unitary-normalized Hecke coefficients at every fixed prime power p^k have the same leading profile as the prime-coefficient average, with the common profile evaluated at the effective coordinate p^k/X under the stated local-uniformity and family-window hypotheses.

Background

The paper studies murmuration, namely coherent biases in Fourier or Hecke coefficients when automorphic objects are ordered by conductor and separated by an arithmetic label such as the root number. Earlier work established the phenomenon for prime coefficients in several settings, while the paper proves an instance for every fixed prime power in the weight-2 level aspect for squarefree levels.

The authors propose a broader universality principle: within any suitably specified self-dual GL(2) family, the same continuous profile governing root-number-weighted prime coefficients should also govern coefficients at p^k when those coefficients are placed at the effective position p^k/X. The conjecture remains unresolved in this general family-level formulation, although the paper verifies the principle for the particular weight-2 squarefree-level family under short-window and growth conditions.