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?

3more comments