This is something which I feel must be out in the literature somewhere, but I have been unable to find anything.
If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A$, then we define the Kummer lattice $K$ to be the minimal primitive sublattice of $H^{2}(\text{Km}(A), \mathbb{Z})$ containing the 16 exceptional curves. The Kummer lattice is negative-definite, even of rank 16 and determinant $2^{6}$. For some details, see page 4 of (https://arxiv.org/pdf/1305.3514.pdf).
The theta function associated to $K$ is of course
$$\theta_{K}(q, \vec{w}) = \sum_{v \in K} q^{v \cdot v} \, \vec{w}^{v}.$$
Has $\theta_{K}(q, \vec{w})$ been studied or described anywhere in the literature? For the life of me, I haven't been able to find a single mention, despite this seeming like a natural question. My guess is that $\theta_{K}(q, \vec{w})$ is some sort of lattice Jacobi form of weight $\frac{16}{2} =8$. It would be great if this were spelled out or computed somewhere.
For what it's worth, I'm actually interested in the generalized theta function with characteristic
$$\theta_{K}^{(a)}(q, \vec{w}) = \sum_{v \in K} q^{(v+\frac{a}{2}) \cdot (v+\frac{a}{2})} \, \vec{w}^{v+ \frac{a}{2}}$$
for some fixed element $a \in K$. If by chance there was also something out there about this more general case that would be great.
