  Let $f(z)=\sum_{n=1}^\infty \lambda_f(n)e^{2\pi i n z}\in
S_{k}^{new}(\Gamma_0(N))$ be a normalized Hecke eigenform of even weight
$k\geq2$ on $\Gamma_0(N)$ without complex multiplication. Let $\mathbb{P}$
denote the set of all primes. We prove that the sequence
$\{\lambda_f(p)\}_{p\in\mathbb{P}}$ does not satisfy Benford's Law in any base
$b\geq2$. However, given a base $b\geq2$ and a string of digits $S$ in base
$b$, the set \[ A_{\lambda_f}(b,S):=\{\text{$p$ prime : the first digits of
$\lambda_f(p)$ in base $b$ are given by $S$}\} \] has logarithmic density equal
to $\log_b(1+S^{-1})$. Thus $\{\lambda_f(p)\}_{p\in\mathbb{P}}$ follows
Benford's Law with respect to logarithmic density. Both results rely on the
now-proven Sato-Tate Conjecture.
