Solution
= Solution
Apply part a in $\mathbb R^2$ to
$$
f_k(x,y)=(x+iy)^ke^{-\pi(x^2+y^2)}.
$$
The supplied identity $\widehat f_k=(-i)^kf_k$ gives
$$
\sum_{\lambda\in\Lambda}\lambda^ke^{-\pi|\lambda|^2}
=(-i)^km(\Lambda)^{-1}
\sum_{\mu\in\Lambda^\vee}\mu^ke^{-\pi|\mu|^2}.
$$
In the notation of the <weighted Gaussian theta sum of a complex lattice>, this is
$$
\theta_k(\Lambda)=(-i)^km(\Lambda)^{-1}\theta_k(\Lambda^\vee).
$$