= Modular transformations of theta characteristics
{title2=$S:(\alpha,\beta)\mapsto(\beta,-\alpha),\quad T:(\alpha,\beta)\mapsto(\alpha,\beta+\alpha-1/2)$}
Translation of tau by one gives $\theta[\alpha,\beta](z,\tau+1)=e^{-\pi i\alpha(\alpha-1)}\theta[\alpha,\beta+\alpha-1/2](z,\tau)$. Gaussian <Poisson summation> gives $\theta[\alpha,\beta](z/\tau,-1/\tau)=\sqrt{-i\tau}\,e^{\pi iz^2/\tau+2\pi i\alpha\beta}\theta[\beta,-\alpha](z,\tau)$. The square root is the holomorphic branch positive at tau=i. These formulas keep the characteristic phase, which is lost if one states only the transformation of unshifted theta constants.
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