  This is a short note generalizing the construction from arXiv:1906.07410,
arXiv:2009.04955 to multi-indices. We recommend to consider both references
first. We obtain polar harmonic Maa{\ss} forms of non-positive integral weight
if the dimension is even and greater than $2$. We provide explicit examples in
dimension $4$, $6$, $8$, and $10$.
