Irreducible polynomials over GF(2) with given trace and subtrace
Let $p(x)$ be a polynomial of degree $n$.
The
trace of $p(x)$ is the coefficient of $x^{n1}$.
The
subtrace of $p(x)$ is the coefficient of $x^{n2}$.
 (trace,subtrace) 
$n$
 (0,0)  (0,1) 
(1,0)  (1,1) 
1 
1  0  1  0


2 
0  0  0  1


3 
0  1  1  0


4 
1  0  1  1


5 
1  2  1  2


6 
2  2  3  2


7 
5  4  4  5


8 
6  8  8  8


9 
15  13  15  13


10 
24  24  24  27


11 
45  48  48  45


12 
85  80  85  85


13 
155  160  155  160


14 
288  288  297  288


15 
550  541  541  550


16 
1008  1024  1024  1024


17 
1935  1920  1935  1920


18 
3626  3626  3626  3654


19 
6885  6912  6912  6885


20 
13107  13056  13107  13107


21 
24940  24989  24940  24989


22 
47616  47616  47709  47616


23 
91225  91136  91136  91225


24 
174590  174760  174760  174760


25 
335626  335462  335626  335462


Enumeration (OEIS)

The number $L(n;k)$ of length $n$ binary Lyndon words of density $k$ is
\[
L(n;k)=\frac{1}{n}\sum_{d\vert \gcd(n,k)} \mu(d)\binom{n/d}{k/d}.
\]
Let $S$ be a subset of $\{0,1,\ldots,n\}$ and let
\[
e(S)=\sum_{k\text{ in }S} L(n;k).
\]

Column (0,0) has value $e(\{k \text{ such that } k+n = 0 \pmod{4}\})$, which is OEIS A042980.

Column (0,1) has value $e(\{k \text{ such that } k+n = 1 \pmod{4}\})$, which is OEIS A042979.

Column (1,0) has value $e(\{k \text{ such that } k+n = 2 \pmod{4}\})$, which is OEIS A042981.

Column (1,1) has value $e(\{k \text{ such that } k+n = 3 \pmod{4}\})$, which is OEIS A042982.
References
 [CMR+03] K. Cattell, C. R. Miers, F. Ruskey, J. Sawada, and M. Serra. The number of irreducible polynomials over GF(2) with given trace and subtrace. J. Combin. Math.
Combin. Comput., 47:31–64, 2003.