So I made a start on learning some genus theory, and found this paper here which has helped me a lot. However, I have found a stumbling block that I would appreciate a clarification on.
This refers to the following lemma which can be found on page 13:
Lemma 4.5. Let $D=-4n$ for a positive integer $n$, and let $f(x_1,x_2)$ be a primitive quadratic form of discriminant $D$. Then the values in $(\Bbb Z/D\Bbb Z)^{\times}$ represented by the principal form of the discriminant $D$ forms a subgroup $H$ of $\text{ker}(\chi)$. The values in $(\Bbb Z/D\Bbb Z)^{\times}$ represented by $f(x_1,x_2)$ form a coset of $H$ in $\text{ker}(\chi)$.
In the proof, the author mentions something along the lines of "using the above identity, it is easy to show that $f(x,y)$ represents $[c]$". However, I'm not too sure as to how one would show something like this.
I would be grateful for either some hints or even the complete solution.
Thanks in advance.