Working in the context of Siegel modular forms, I am interested about whether any closed form is known for the sum $$f(Z) = \sum_{T \ge 0} e^{2\pi i \mathrm{tr}(TZ)},$$ taken over symmetric positive semidefinite matrices $T$ with integer diagonal and half-integer off-diagonal entries, where $Z$ is a symmetric matrix with positive-definite imaginary part.
Even the case of $(2 \times 2)$ matrices, which simplifies to the sum $$f(z_1,z_2,z_3) = \sum_{s,u,su - t^2/4 \ge 0} p^s q^t r^u, \; p = e^{2\pi i z_1} \; q = e^{2\pi i z_2} \; r = e^{2\pi i z_3}$$ seems rather difficult.