= Character theta transformation
{title2=$\Theta_\chi(\tau)=\dfrac{g(\chi)}{i^aN^{a+1}(-i\tau)^{a+1/2}}\Theta_{\overline\chi}(-1/(N^2\tau))$}
For a <primitive Dirichlet character> modulo $N>1$ of parity $a\in\{0,1\}$, put $\Theta_\chi(\tau)=\sum_n n^a\chi(n)e^{\pi i n^2\tau}$, extending the character by zero on nonunits. The square root is analytic with positive value when $-i\tau$ is positive real. Apply <Poisson summation> separately in each residue class modulo $N$, and use the <finite Fourier transform of a primitive Dirichlet character>. The <Fourier transforms> of $e^{\pi i\tau x^2}$ and $x e^{\pi i\tau x^2}$ are $(-i\tau)^{-1/2}e^{-\pi i t^2/\tau}$ and $(t/\tau)(-i\tau)^{-1/2}e^{-\pi i t^2/\tau}$. These give the formula, including its odd-character phase.
Back to article page
