Fix $K$ a number field and a modulus $\mathfrak m.$ Theory around the class number formula tells us that primes $\mathfrak p$ of $K$ equidistribute over all ideal classes in the normal class group, $J_K^1/P_K^{1,1}.$ (Here we use Dirichlet density.) Is it in general true that primes equidistribute over ideal classes in $J_K^{\mathfrak m}/P_K^{\mathfrak m,1}$?
Additionally, is there a corresponding "ray class number formula"? I am aware of the typical ray class number formula in terms of the class number (say, Milne V.1.7) but am interested if there's a residue of some $\zeta$-function around here.
After doing some thinking, I realized the answer is "yes" for somewhat unsatisfying reasons. Simply apply Chebotarev to the needed ray class field to get the result.