Consider an $ [\![n,k]\!] $ non-stabilizer code. Define the weight enumerator polynomial $ A(x) $ of the code as
$$
A(x):=A_0+A_1x+\dots+A_nx^n
$$  
where 
$$
A_j:=\frac{1}{(2^k)^2} \sum_{p \in P_n,\,\mathrm{wt}(p)=j} |\mathrm{tr}(p \Pi)|^2.
$$
Here $ \Pi $ is the projector onto the code subspace. 
For stabilizer codes the $ A_j $ are always integers see 
https://quantumcomputing.stackexchange.com/questions/30184/are-the-coefficients-of-the-weight-enumerator-polynomial-of-a-stabilizer-code-al/30187#30187 . Is it the case that the $ A_j $ are always integers for on stabilizer codes as well?

If not, what is an example of a non-stabilizer code with $ A_j $ not integers?




Thoughts on a counterexample:

The examples of non-stabilizer codes given here
 
https://quantumcomputing.stackexchange.com/questions/27610/example-non-stabilizer-code

all seem to have logical code words which are uniform signed ($ \pm1 $) superposition of computational basis kets (times a global normalization). My intuition is that a code of that type will always have $ A_j $ integer. So perhaps the hunt for a counterexample would start with finding a non-stabilizer code which is truly exotic in the sense that the code words are not just signed superposition of computational basis kets.