  In 1998 Friedlander and Iwaniec proved that there are infinitely many primes
of the form $a^2+b^4$. To show this they defined the spin of Gaussian integers
by the Jacobi symbol, and one of the key ingredients in the proof was to show
that the spin becomes equidistributed along Gaussian primes. To generalize this
we define the cubic spin of ideals of
$\mathbb{Z}[\zeta_{12}]=\mathbb{Z}[\zeta_3,i]$ by using the cubic residue
character on the Eisenstein integers $\mathbb{Z}[\zeta_3]$. Our main theorem
says that the cubic spin is equidistributed along prime ideals of
$\mathbb{Z}[\zeta_{12}]$. The proof of this follows closely along the lines of
Friedlander and Iwaniec. The main new feature in our case is the infinite unit
group, which means that we need to show that the definition of the cubic spin
on the ring of integers lifts to a well-defined function on the ideals. We also
explain how the cubic spin arises if we consider primes of the form $a^2+b^6$
on the Eisenstein integers.
