Weighted Gaussian theta sum of a complex lattice
= Weighted Gaussian theta sum of a complex lattice
{title2=$\theta_k(\Lambda)$}
For an even nonnegative integer $k$,
$$
\theta_k(\Lambda)=\sum_{\lambda\in\Lambda}\lambda^ke^{-\pi|\lambda|^2}.
$$
Poisson summation and the Fourier eigenfunction identity for $(x+iy)^ke^{-\pi(x^2+y^2)}$ give
$$
\theta_k(\Lambda)=(-i)^k m(\Lambda)^{-1}\theta_k(\Lambda^\vee).
$$