Almost all primes satisfy the Atkin–Serre conjecture and are not extremal

Ayla Gafni, Jesse Thorner, Peng Jie Wong

Research output: Contribution to journalArticlepeer-review


Let f(z)=∑n=1∞af(n)e2πinz be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight k≥ 2. Deligne’s proof of the Weil conjectures shows that |af(p)|≤2pk-12 for all primes p. We prove for 100% of primes p that 2pk-12loglogp/logp<|af(p)|<⌊2pk-12⌋. Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin–Serre conjecture is satisfied for 100% of primes, and the upper bound shows that | af(p) | is as large as possible (i.e., p is extremal for f) for 0% of primes. Our proofs use the effective form of the Sato–Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of f due to Newton and Thorne.

Original languageEnglish (US)
Article number31
JournalResearch in Number Theory
Issue number2
StatePublished - Jun 2021

ASJC Scopus subject areas

  • Algebra and Number Theory


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