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 Are the coefficients of the weight enumerator polynomial of a stabilizer code always integers? . 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

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.