Theta-constant inversion and translation laws (${\Theta }_2,{\Theta }_3,{\Theta }_4$)

In the $e^{\pi iz}$ convention, Poisson summation gives ${\Theta }_3(-1/z)=\sqrt{-iz}{\Theta }_3(z)$ and interchanges ${\Theta }_2,{\Theta }_4$ with the same factor. Translation by one sends ${\Theta }_2$ to $e^{\pi i/4}{\Theta }_2$ and interchanges ${\Theta }_3,{\Theta }_4$. Eighth powers remove every phase and square-root ambiguity when proving integral-weight modularity.