I am reading section 1.2 of A First Course in Modular Forms. Let $q=e^{2\pi i\tau}$, where $\tau\in\cal H$ is assumed to be in the upper half plane, and define $\theta(\tau)=\sum_{n\in\Bbb Z}q^{n^2}$. Clearly $\theta(\tau+1)=\theta(\tau)$. The text states we will use Poisson summation to prove $\theta(-\frac{1}{4\tau})=\sqrt{-2i\tau}\theta(\tau)$ (using the principal branch of $\sqrt{}$) later in the book. Let ${\rm GL}_2(\Bbb Z)$ act on $\cal H$ by $[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}]\tau=\frac{a\tau+b}{c\tau+d}$. Even though we can normalize $[\begin{smallmatrix}0&-1\\4&0\end{smallmatrix}]$ to be in ${\rm SL}_2(\Bbb Z)$ the relation is not consonant with the modular form transformation law, since $[-2i~0]$ is not the bottom row. So the book says to right-conjugate $[\begin{smallmatrix}1&-1\\0&1\end{smallmatrix}]$ by $[\begin{smallmatrix}0&-1\\4&0\end{smallmatrix}]$ and then compute
$$\theta([\begin{smallmatrix}1&0\\4&1\end{smallmatrix}]\tau)=\theta([\begin{smallmatrix}0&-1\\4&0\end{smallmatrix}]^{-1}[\begin{smallmatrix}1&-1\\0&1\end{smallmatrix}][\begin{smallmatrix}0&-1\\4&0\end{smallmatrix}]\tau)=\sqrt{1+4\tau}\theta(\tau)$$
using the already-known relations. Note the computation uses the fact that $[\begin{smallmatrix}0&-1\\4&0\end{smallmatrix}]^{-1}$ and $[\begin{smallmatrix}0&-1\\4&0\end{smallmatrix}]$ have the same action, as the former is a scalar multiple of the latter and scalar matrices act trivially. Therefore this successfully obtains the form $\theta([\begin{smallmatrix}a&b\\c&d\end{smallmatrix}]\tau)=(c\tau+d)^k\theta(\tau)$ for some $[\begin{smallmatrix}a&b\\c&d\end{smallmatrix}]\in{\rm SL}_2(\Bbb Z)$, $k\in\Bbb Q$.
My question is, what is the motivation for the conjugation? Obviously it works, but I'd never have thought of it on my own, so I'm wondering what the "source" for this trick is. Or, could one reason one's way to this conjugation as the correct route to go from scratch?