I am working on the imaginary quadratic field $K=\mathbb{Q} (w)$ where $w = \frac{-1 + \sqrt{-3}}{2}$. I need to find the density of primes $\pi \equiv 1 (9)$ in ray class field $J_K^m / P_K^m$ (so I will take m=9). I want to use a refined version of generalized Chebotarev Density Theorem and to do this, I need a relation between the Galois group $G(K^m/K)$ and the ray class group $J_K^m/P_K^m$. In Neukirch's algebraic number theory book, Theorem 7.1 gives that there exists a surjective homomorphism $ CL_K^m = J_K^m/P_K^m \rightarrow G(L/K)$ for an abelian extension $L/K$ with kernel $(N_{L/K}J_L^m)P_K^m$.
Before this, (below definition 6.2) it is stated that the Galois group of the ray class field is canonically isomorphic to the ray class group $\mod m$, so $G(K^m/K) \cong C_K/ C_K^m $. Also we have an isomorphism $C_K/C_K^m \cong CL_K^m$ (by Proposition 1.9). Then what I understand is that the surjective homomorphism of Theorem 7.1 $ CL_K^m = J_K^m/P_K^m \rightarrow G(L/K)$ has to be a an isomorphism when we take $L=K^m$ so that its kernel has to be trivial, namely, it has to be $P_K^m$. So in this case $(N_{L/K}J_L^m)$ has to be $1$. Since this is an isomorphism, the size of $P_K^m$ equals to the size of the identity automorphism of G(L/K). In this way I catch those primes $\pi \equiv 1 (9)$ since those are the elements in $P_K^m$ by definition. I am wondering whether what I understand is correct up to now. Thank you in advance for any help!