Summary: Let \(f(z)=\sum_{n=1}^{\infty} a_f(n)e^{2\pi inz}\) be a non-CM holomorphic cuspidal newform of trivial nebentypus and even integral weight \(k\geq 2\). Deligne’s proof of the Weil conjectures shows that \(|a_f(p)|\le 2p^{\frac{k-1}{2}}\) for all primes \(p\). We prove for 100% of primes \(p\) that \(2p^{\frac{k-1}{2}}{\log \log p}/{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor\). 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 \(|a_f(p)|\) 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.