Theta-constant inversion and translation laws
= Theta-constant inversion and translation laws
{title2=$\Theta_2,\Theta_3,\Theta_4$}
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.