I am interested in considering examples of abelian varieties that are principally polarized with CM in dimension three. However, I am struggling to construct or find even a single instance.

In particular, I attempted to take a totally real field of degree 3 (with relatively small discriminant) from LMFDB, construct the corresponding CM field of degree 6, then considering the Hermitian form $(a,b) \mapsto \textrm{Tr} (a\overline{b})$. When restricting to get the Riemann form, I discover that the degree of the corresponding polarizations are powers of the discriminant of the field (probably not an accident). As far as being principally polarized though, this is not helpful.

Searching the literature, I've only found the following result: https://arxiv.org/pdf/1208.5599.pdf which only discusses even dimensions.

I am wondering if there is a reference that might discuss either odd dimension or just $d = 3$.

Edit: I forgot an important criterion that I'd like the variety to have a compatible cyclic action of degree $3$ or degree $4$ (I've focused mainly on degree $3$ so far).