Local weights in the trace formula and the q = 2 anomaly in the dilation law of murmurations

Wachs observed that the covariance Cov(a_p, a_q) of rank-0 elliptic curves ordered by conductor follows a dilation law that fails for q = 2, and left the discrepancy unexplained. We explain it at the level of the trace formula for weight-2 newforms: the composite index qp produces the same terms as a prime index, each multiplied by an explicit local weight at q (3/4, 3 and 3/2 for q = 2), computed exactly and proved for the index qp. Under an explicit transfer hypothesis, the reweighted dilation describes the elliptic data in and out of sample, up to conductors 5·10^5, and reproduces Wachs’s value 0.508 on his own subpopulation, raising it to 0.892. For curves ordered by height we prove the conditional structure of smoothed murmurations. The archive contains the paper and the code that reproduces every table and figure (code under the MIT licence). The work was carried out with substantial assistance from Claude (Anthropic), as stated in the paper.