We use the definitions of this question.
Is the following proposition true? If yes, how do we prove it?
Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). Let $\chi\colon (\mathbb{Z}/D\mathbb{Z})^\times\rightarrow \mathbb{Z}^\times = \{-1, 1\}$ be the homomorphism defined in the proposition 2 of this question.
Let $F = ax^2 + bxy + cy^2$ be a primitive form of discriminant $D$. If $D < 0$, we assume that $F$ is positive definite. Let $H = \{ [m] \in (\mathbb{Z}/D\mathbb{Z})^\times$; $m$ is represented by the principal form of discriminant $D$(for the definition of the principal form, please see this question)$\}$.
Let $S = \{ [m] \in (\mathbb{Z}/D\mathbb{Z})^\times; m$ is represented by $F\}$.
Then $S$ is a coset of $H$ in Ker$(\chi)$.
