Irreducible polynomials over $GF(4)$ with given trace and subtrace

Let $p(x)$ be a polynomial of degree $n$. The trace of $p(x)$ is the coefficient of $x^{n-1}$. The subtrace of $p(x)$ is the coefficient of $x^{n-2}$. Below we use $a=\operatorname{root}(x^2+x+1)$ and $b=1+a$.

(trace,subtrace)
$n$ (0,0) (0,1)
(0,$a$)
(0,$b$) (1,0)
($a$,0)
($b$,0) (1,1)
($a$,$b$)
($b$,$a$) (1,$a$)
(1,$b$)
($a$,1)
($a$,$a$)
($b$,1)
($b$,$b$)
1 1 0 1 0 0
2 0 0 0 0 1
3 2 1 1 2 1
4 0 4 4 4 4
5 15 12 15 12 12
6 40 40 40 40 45
7 153 144 144 153 144
8 480 512 512 512 512
9 1841 1813 1841 1813 1813
10 6528 6528 6528 6528 6579
11 23901 23808 23808 23901 23808
12 87040 87380 87380 87380 87380
13 322875 322560 322875 322560 322560
14 1198080 1198080 1198080 1198080 1198665
15 4474738 4473647 4473647 4474738 4473647
16 16773120 16777216 16777216 16777216 16777216

	(trace,subtrace)
$n$	(0,0)	(0,1) (0,$a$) (0,$b$)	(1,0) ($a$,0) ($b$,0)	(1,1) ($a$,$b$) ($b$,$a$)	(1,$a$) (1,$b$) ($a$,1) ($a$,$a$) ($b$,1) ($b$,$b$)
1	1	0	1	0	0
2	0	0	0	0	1
3	2	1	1	2	1
4	0	4	4	4	4
5	15	12	15	12	12
6	40	40	40	40	45
7	153	144	144	153	144
8	480	512	512	512	512
9	1841	1813	1841	1813	1813
10	6528	6528	6528	6528	6579
11	23901	23808	23808	23901	23808
12	87040	87380	87380	87380	87380
13	322875	322560	322875	322560	322560
14	1198080	1198080	1198080	1198080	1198665
15	4474738	4473647	4473647	4474738	4473647
16	16773120	16777216	16777216	16777216	16777216

Examples

Monic irreducible polynomials over $GF(4)$ of degree $n=2$. Let $r =\operatorname{root}(x^2+x+a)$. Then $r^4=r+1$.

(trace,subtrace)
(1,$a$) $x^2+x+a=(x+r)(x+r+1)$
(1,$b$) $x^2+x+b=(x+r+a)(x+r+b)$
($a$,1) $x^2+ax+1=(x+ar)(x+ar+a)$
($a$,$a$) $x^2+ax+a=(x+ar+1)(x+ar+b)$
($b$,1) $x^2+bx+1=(x+br+1)(x+br+a)$
($b$,$b$) $x^2+bx+b=(x+br)(x+br+b)$

(trace,subtrace)
(1,$a$)	$x^2+x+a=(x+r)(x+r+1)$
(1,$b$)	$x^2+x+b=(x+r+a)(x+r+b)$
($a$,1)	$x^2+ax+1=(x+ar)(x+ar+a)$
($a$,$a$)	$x^2+ax+a=(x+ar+1)(x+ar+b)$
($b$,1)	$x^2+bx+1=(x+br+1)(x+br+a)$
($b$,$b$)	$x^2+bx+b=(x+br)(x+br+b)$

Monic irreducible polynomials over GF(4) of degree $n=3$.

(trace,subtrace)
(0,0) $x^3+a$ $x^3+b$
(0,1) $x^3+x+1$
(0,$a$) $x^3+ax+1$
(0,$b$) $x^3+bx+1$
(1,0) $x^3+x^2+1$
(1,1) $x^3+x^2+x+a$ $x^3+x^2+x+b$
(1,$a$) $x^3+x^2+ax+b$
(1,$b$) $x^3+x^2+bx+a$
($a$,0) $x^3+ax^2+1$
($a$,1) $x^3+ax^2+x+b$
($a$,$a$) $x^3+ax^2+ax+a$
($a$,$b$) $x^3+ax^2+bx+a$ $x^3+ax^2+bx+b$
($b$,0) $x^3+bx^2+1$
($b$,1) $x^3+bx^2+x+a$
($b$,$a$) $x^3+bx^2+ax+a$ $x^3+bx^2+ax+b$
($b$,$b$) $x^3+bx^2+bx+b$

(trace,subtrace)
(0,0)	$x^3+a$	$x^3+b$
(0,1)	$x^3+x+1$
(0,$a$)	$x^3+ax+1$
(0,$b$)	$x^3+bx+1$
(1,0)	$x^3+x^2+1$
(1,1)	$x^3+x^2+x+a$	$x^3+x^2+x+b$
(1,$a$)	$x^3+x^2+ax+b$
(1,$b$)	$x^3+x^2+bx+a$
($a$,0)	$x^3+ax^2+1$
($a$,1)	$x^3+ax^2+x+b$
($a$,$a$)	$x^3+ax^2+ax+a$
($a$,$b$)	$x^3+ax^2+bx+a$	$x^3+ax^2+bx+b$
($b$,0)	$x^3+bx^2+1$
($b$,1)	$x^3+bx^2+x+a$
($b$,$a$)	$x^3+bx^2+ax+a$	$x^3+bx^2+ax+b$
($b$,$b$)	$x^3+bx^2+bx+b$

Enumeration (OEIS)

Column (0,0) is OEIS A074031.
Column (0,1),(0,$a$),(0,$b$) is OEIS A074032.
Column (1,0),($a$,0),($b$,0) is OEIS A074033.
Column (1,1),($a$,$b$),($b$,$a$) is OEIS A074034.
Column (1,$a$),(1,$b$),($a$,1),($a$,$a$),($b$,1),($b$,$b$) is OEIS A074035.

Data source: Entries for $n=1,2,...,9$ were done by exhaustive Maple program (also $n=10$ entries (0,$x$)). For $n$ odd entries match corresponding entries on the Lyndon words over $GF(4)$ page via a theorem of Miers and Ruskey. Some of the others are educated guesses.