We define the sixth power character on $Z[w]$, where $w=\frac{-1+\sqrt{-3}}{2}$, the cubic root of unity, like follows:
Let $u,v \in Z[w]$, such that $u$ is a prime in $Z[w]$, and $(v,u)=1$, then exists a only one integer m such that, $v^{\frac{Nu-1}{6}}\equiv \zeta_{6}^{m} (mod\ u).$
We define $( \frac{v}{u})_{6} = \zeta _{6}^{m}$ like above.
Now, $3=-w^{2}(1-w)^{2}$, and let be $u$ a primary prime in $Z[w]$, $u\neq 2$.
Does exist a criteria to compute $(\frac{1-w}{u})_{6}$, which depents only on the residue class of $u$ modulo some rational integer $n$?
What is that $n$? What is the criteria?
