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

Let $$f(z)=\sum _{n=1}^{\infty } a_f(n)e^{2\pi i n z}$$ be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight $$k\ge 2$$ . Deligne’s proof the Weil conjectures shows that $$|a_f(p)|\le 2p^{\frac{k-1}{2}}$$ for all primes p. We prove 100% p $$ 2p^{\frac{k-1}{2}}{\log \log p}/{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor .$$ Our gives an effective upper bound size exceptional set. The lower Atkin–Serre conjecture is satisfied primes, $$|a_f(p)|$$ as large possible (i.e., extremal f) 0% primes. proofs use form Sato–Tate proved by second author, which relies on recent automorphy symmetric powers f due to Newton Thorne.