Modular transformations of theta characteristics ($S:(\alpha ,\beta )\mapsto (\beta ,-\alpha ),T:(\alpha ,\beta )\mapsto (\alpha ,\beta +\alpha -1/2)$)

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 $\theta [\alpha ,\beta ](z/\tau ,-1/\tau )=\sqrt{-i\tau }{e}^{\pi i{z}^2/\tau +2\pi i\alpha \beta }\theta [\beta ,-\alpha ](z,\tau )$. 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.