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PreprintCC BY 4.0arXiv

Universal murmuration and Hecke augmentation

  • Shenghao Hua
  • Chung Pang Mok
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Abstract

Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels. We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position pk/X, where X is the conductor scale. For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed k under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and Müller. Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.

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Cite

BibTeXRIS
BibTeX
@misc{hua2026universal,
  title = {{Universal murmuration and Hecke augmentation}},
  author = {Shenghao Hua and Chung Pang Mok},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00777},
  eprint = {2609.00777},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00777},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Universal murmuration and Hecke augmentation
AU  - Shenghao Hua
AU  - Chung Pang Mok
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00777
UR  - https://arxiv.org/abs/2609.00777
AB  - Prime coefficients of elliptic curves exhibit murmurations, statistical patterns that support the prediction of arithmetic labels. We conjecture that root number weighted averages of unitary normalized coefficients at primes and prime powers sample the same leading profile when placed at the effective position p^k/X, where X is the conductor scale. For weight $2$ newforms of squarefree level in the level aspect, we prove this principle for every fixed k under suitable short-window and growth conditions, extending Zubrilina's prime case and the square-case analysis of Kundu and Müller. Experiments with elliptic curve isogeny class representatives show that the resulting prime power features improve root number prediction and give a smaller gain in distinguishing ranks $0$ and $1$.
ER  -