Given a polynomial $a(t)\in F_2[t]$, for which irreducible $f(t)\in F_2[t]$ does the trace of $a(t)\bmod f(t)$ equal 0/1? By Artin reciprocity applied to the extension defined by $x^2-x-a(t)$ it should depend only on the residue of $f(t)$ modulo some conductor, but what does this reciprocity law look like explicitly for this case?