# Relation between a primitive binary quadratic form of discriminant $D$ and a certain Dirichlet character modulo $D$.

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$(\chi)$.

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Excuse me, what exactly do you mean by a residue class?I cannot find the definition in the quoted links. Thanks. –  awllower May 14 '13 at 15:58
@awllower I edited the question. –  Makoto Kato May 15 '13 at 9:30
Thanks for responding then. :) –  awllower May 15 '13 at 14:06