dg.differential geometry – almost complex \$mathbb{Z}^{6}\$-action

Suppose we take an almost complex structure on $$mathbb{T}^{6}$$ with $$c_{1} neq 0$$ (there should be infinitely many homotopy classes satisfying this requirement). Now pull it back to the universal cover $$mathbb{R}^6$$, giving an almost complex structure $$J$$, it should be invariant by the deck group action of $$mathbb{Z}^{6}$$.

My question is there a case where we can explicitly describe $$J$$ and the group action of $$mathbb{Z}^6$$ in coordinates?

(I asked this question in stack exchange in August https://math.stackexchange.com/questions/3789163/almost-complex-actions-on-mathbbr6 with no answer so I thought I would ask here.)