I am trying to show that
the action of PSL$_2(\mathbb Z)$ on a quadratic form by $g \cdot Q = Q(ax + by, cx + dy)$ preserves the set of properly represented numbers, where $$g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$
Clearly $g$ is a Mobius transformation.
I have already proved that this action is a right action $(g \circ(h \circ Q)) = (gh) \circ Q)$ and that it preserves the discriminant of $Q$.
Can anyone point me in the right direction? My current idea is to use the fact that the smallest three values a reduced BQF will give are $A$, $C$, and $A + C - |B|$. If two BQFs both have these as their smallest three, they must properly represent the same numbers.