Singular values of principal moduli

Let g be a principal modulus with rational Fourier coefficients for a discrete subgroup of SL2(R) lying in between Γ(N) and Γ0(N)† for a positive integer N. Let K be an imaginary quadratic field. We introduce a relatively simple proof, without using Shimuraʼs canonical model, of the fact that the singular value of g generates the ray class field modulo N or the ring class field of the order of conductor N over K. Further, we construct a primitive generator of the ray class field Kc of arbitrary modulus c over K from Hasseʼs two generators.