  We state and prove an identity which connects theta series associated with
binary quadratic forms of idoneal discriminants $\Delta$ and $\Delta p^2$, for
$p$ a prime. Employing this identity, we extend the results of Toh by writing
the theta series of forms of discriminant $\Delta p^2$ as a linear combination
of Lambert series. We then use these Lambert series decompositions to give
explicit representation formulas for the forms of discriminant $\Delta p^2$.
Lastly, we give a generalization of our main identity, which employs a map of
Buell to connect forms of discriminant $\Delta$ to $\Delta p^2$. Our
generalized identity links theta series associated with a single form of
discriminant $\Delta$ to a theta series associated with forms of discriminant
$\Delta p^2$, where $\Delta$ and $\Delta p^2$ are no longer required to be
idoneal.
