Benford's Law for coefficients of newforms

Let f(z) = Sigma(infinity)(n=1) lambda(f) (n)e(2 pi inz) is an element of S-k(new) (Gamma(0)(N)) be a newform of even weight k >= 2 on Gamma(0)(N) without complex multiplication. Let P denote the set of all primes. We prove that the sequence {lambda(f) (p)}(p is an element of P) does not satisfy Benford's Law in any integer base b >= 2. However, given a base b >= 2 and a string of digits S in base b, the set A(lambda f) (b, S) := {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 is an element of P) follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.