Gauss sum variant: $ \sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)} \big) $

Question

Motivated by an example in Chern Simons theory, let $p \in \mathbb{Z}$ be prime, can anyone compute this sum:

$$ \sum_{ a, b \in \mathbb{Z} } e^{-i \pi (a^2+b^2)/p } \big( e^{\pi(a-b)} + e^{-\pi(a-b)} \big) $$

This looks like the Gauss sum, except it is multiplied by something that is not periodic so we sum over $a, b \in \mathbb{Z}$.