  This article covers three topics. (1) It establishes links between the
density of certain subsets of the set of primes and related subsets of the set
of natural numbers. (2) It extends previous results on a conjecture of Bruinier
and Kohnen in three ways: the CM-case is included; under the assumption of the
same error term as in previous work one obtains the result in terms of natural
density instead of Dedekind-Dirichlet density; the latter type of density can
already be achieved by an error term like in the prime number theorem. (3) It
also provides a complete proof of Sato-Tate equidistribution for CM modular
forms with an error term similar to that in the prime number theorem.
