Prime factorization over the Eisenstein integers $mathbb{Z}[zeta]$

I am trying to write a function f[a_,b_] which takes in two integers $a,b$ and returns the unique factorization of $a+be^{2pi i/3}$ into the primes belonging to $mathbb{Z}[e^{2pi i/3}]$.

In other words, f gives the factorization of $a+be^{2pi i /3}$ into Eisenstein primes.

Useful facts. The Eisenstein primes have the following possible forms:

• $1-e^{2pi i/3}$
• Primes in $mathbb{Z}$ congruent to $2 pmod 3$
• The Eisenstein integers $g,bar{g}$ such that $g bar{g} = q$, where $q equiv 1 pmod 3$ is prime in $mathbb{Z}$

Here, I denote $bar{z}$ to mean $(a+bB)(a+bB^2)$. Also, Eisenstein integers are the elements of $mathbb{Z}[e^{2 pi i/3}]$.

Desired output. I am trying to create f to function similar to FactorInteger. It would output a list of pairs {x,y} for each Eisenstein prime $x$ dividing $v=a+be^{2pi i/3}$, with $y$ being the highest power of $x$ dividing $v$.

Some convention. I denote B=1-E^(2Pi*i/3), and I want f to use this instead of E^(2Pi*I/3).

Example output. Because $24-36B=B^2*2^2*(5+2B)$, we have f[24,-36] = { {B,2} , {2,2} , {5+2B,1} }.

My ‘work’. There was stuff here, but I realized I was forgetting something. Currently working on the program in question.