nt.number theory – Is there a Siegel-Weil formula for generalized Siegel theta functions?

For the usual Siegel-Narain theta functions $Theta(m,tau)$, the Siegel-Weil formula relates the integral of it on the 2D dimensional Narain moduli space to the Eisenstein Series on $SL(2,Z)$ as $E_{D/2}(tau)$.
To be more concrete

$int _{{mathcal{M}}} dmu Theta(m,tau)=E_{D/2}(tau)$

where m denotes a point in the moduli space.

I wonder if there is an analogous formula for the generalized Siegel theta functions $Theta (alpha,beta)(m,tau)$?

where $Theta (alpha,beta)(m,tau)$ means that we do some shifts in the lattice sum by vectors related to $alpha$ and $beta$. (twisted partition function for the compact boson in physics language)

Thanks for your help!

(sorry I am a physicist trying to understand these things, so might be using non-standard terminologies in math)