I'm trying to generate the set of solutions of a specific diophantine equation over Z[i]. The equation is the following:

$$ z_1^2 + z_2^2 + z_1*z_2 + 39 = 0$$

with $ z_1$ (resp $z_2$) such that $\exists a,b \in \mathbb{Z} , z_1$ (resp $z_2$) $= a + ib $

Is there a specific tehnique to deal with Gaussian Integers in Diophantine equations ?

Regards