Siegel's bad character

Question

Let $K$ be an imaginary quadratic field with discriminant $d_K$. Suppose that $d_K=gt$, where either $g,t$ are discriminants or have the value $g=1,t=d$. Let $f$ be an additional discriminant of a quadratic field, or $f=1$ and at the same time $g\neq1$. Denote by $G$ the positive discriminant from $fg,ft$, and $T$ the negative one. For a prime $\mathfrak p$ of $K$ such that $(\mathfrak p,f)=1$ let
$$\chi(\mathfrak p)=\begin{cases}\left(\frac{G}{N(\mathfrak p)}\right) \text{ if }\left(\frac{G}{N(\mathfrak p)}\right)\neq 0,\\ \left(\frac{T}{N(\mathfrak p)}\right) \text{ if } \left(\frac{T}{N(\mathfrak p)}\right)\neq 0.\end{cases}$$ Siegel writes that

"Für beliebige zu $f$ teilerfremde Ideale a wird $\chi(\mathfrak a)$ auf multiplikative Art gebildet und erweist sich dann als eigentlicher Charakter der Gruppe der Ringklassen mit dem Führer $\lvert f\rvert$."

What exactly he means by "Gruppe der Ringklassen mit dem Führer $f$"? Is it the class group of the order with conductor $f$? What is more systematic or modern way of viewing these characters (and the "Ringklassen")?

In other words, how to compute the weights in the following formula

$$\varepsilon_G^{h_Gh_T}=\prod_{C\in \text{Cl}_f}h(\tau_C)^{-\chi(C)},$$ where $$h(\tau)=y^{1/2}\lvert \eta(\tau)^2\rvert, \\ \tau=x+iy,\\ \tau_C =\text{CM point representing the class $C$},\\\varepsilon_G = \text{the fundamental unit in $\mathbb Q(G^{1/2})$}.$$ See also my previous questions. According to these, in each class of $\mathcal O_f$ is an ideal of norm equal to $1$. But then is the character above trivial?

Answer 1

The ring class group is a special kind of a ray class group, introduced connection with the theory of complex multiplication in complex quadratic number fields. Given an integer $f > 1$, consider the group $D_f$ of all ideals coprime to $f$. The group $P_f$ of principal ideals is the group generated by ideals $(\alpha) \in D_f$ with $\alpha \equiv z \bmod f$ for some integer $z$. The ring class group modulo $f$ is simply the quotient $D_f/P_f$, and the ring class field defined modulo $f$ is obviously a subextension of the ray class field modulo $f$. In Siegel's definition of the quadratic character, ${\mathfrak p}$ is an ideal in the maximal order, not in the order with conductor $f$. It is true, however, that the ring class group modulo $f$ is isomorphic to the ideal class group of the order with conductor $f$, but the isomorphism is not as natural as one might expect. You should definitely look into the books by Cox and Cohn that I have already recommended.

Computing the corresponding CM-points is usually done via quadratic forms: to $Q(x,y) = Ax^2 + Bxy + Cy^2$ you associate the root of the equation $Q(z,1) = 0$ in the upper half plane.