Elements of $GF(7^n)$ with given trace and subtrace

If $a$ is an element of $GF(7^n)$, then the (absolute) trace of $a$ is \[ \operatorname{tr}(a)=a+a^7+a^{49}+\cdots+a^{7^{n-1}}. \] Alternatively, we could define $\operatorname{tr}(a)$ to be the negation of the coefficient of $x^{n-1}$ in the (characteristic) polynomial \[ p(x) = (x - a) (x - a^7) (x - a^{49}) \cdots (x - a^{7^{n-1}}). \] The subtrace of $a$ is the coefficient of $x^{n-2}$ in $p(x)$. The coefficients of $p(x)$ are guaranteed to be elements of $GF(7)$, so the trace and subtrace are elements of $GF(7)$ (i.e., the value is $0,1,\ldots,6$).

(trace,subtrace)
$n$ (0,0) (0,1)
(0,2)
(0,4) (0,3)
(0,5)
(0,6) (1,0)
(2,0)
(3,0)
(4,0)
(5,0)
(6,0) (1,1)
(2,4)
(3,2)
(4,2)
(5,4)
(6,1) (1,2)
(2,1)
(3,4)
(4,4)
(5,1)
(6,2) (1,3)
(2,5)
(3,6)
(4,6)
(5,5)
(6,3) (1,4)
(2,2)
(3,1)
(4,1)
(5,2)
(6,4) (1,5)
(2,6)
(3,3)
(4,3)
(5,6)
(6,5) (1,6)
(2,3)
(3,5)
(4,5)
(5,3)
(6,6)
1 1 0 0 1 0 0 0 0 0 0
2 1 2 0 0 0 1 2 2 0 2
3 13 6 6 6 6 6 6 6 13 6
4 49 42 56 42 56 56 49 42 42 56
5 301 350 350 350 350 350 350 350 350 301
6 2401 2352 2450 2450 2401 2352 2352 2450 2352 2450
7 16807 17150 16464 16807 16807 16807 16807 16807 16807 16807

	(trace,subtrace)
$n$	(0,0)	(0,1) (0,2) (0,4)	(0,3) (0,5) (0,6)	(1,0) (2,0) (3,0) (4,0) (5,0) (6,0)	(1,1) (2,4) (3,2) (4,2) (5,4) (6,1)	(1,2) (2,1) (3,4) (4,4) (5,1) (6,2)	(1,3) (2,5) (3,6) (4,6) (5,5) (6,3)	(1,4) (2,2) (3,1) (4,1) (5,2) (6,4)	(1,5) (2,6) (3,3) (4,3) (5,6) (6,5)	(1,6) (2,3) (3,5) (4,5) (5,3) (6,6)
1	1	0	0	1	0	0	0	0	0	0
2	1	2	0	0	0	1	2	2	0	2
3	13	6	6	6	6	6	6	6	13	6
4	49	42	56	42	56	56	49	42	42	56
5	301	350	350	350	350	350	350	350	350	301
6	2401	2352	2450	2450	2401	2352	2352	2450	2352	2450
7	16807	17150	16464	16807	16807	16807	16807	16807	16807	16807

Examples

Let $GF(7^2)$ be defined by the field extension $GF(7)[x]/(3+5x+x^2)$. The one element of $GF(7^2)$ with trace 3 and subtrace 4 is $\{5\}$.
Let $GF(7^2)$ be defined by the field extension $GF(7)[x]/(3+5x+x^2)$. The two elements of $GF(7^2)$ with trace 5 and subtrace 2 are $\{4+2x, 1+5x\}$.
Let $GF(7^3)$ be defined by the field extension $GF(7)[x]/(3+x^2+x^3)$. The six elements of $GF(7^3)$ with trace 1 and subtrace 6 are $\{1+2x, 2+5x, 1+3x+x^2, 4+5x+x^2, 6+2x+6x^2, 2+4x+6x^2\}$.

Enumeration (OEIS)

Column (0,0) is OEIS A074014.
Column (0,1),(0,2),(0,4) is OEIS A074015.
Column (0,3),(0,5),(0,6) is OEIS A074016.
Column (1,0),(2,0),(3,0),(4,0),(5,0),(6,0) is OEIS A074017.
Column (1,1),(2,4),(3,2),(4,2),(5,4),(6,1) is OEIS A074018.
Column (1,2),(2,1),(3,4),(4,4),(5,1),(6,2) is OEIS A074019.
Column (1,3),(2,5),(3,6),(4,6),(5,5),(6,3) is OEIS A074020.
Column (1,4),(2,2),(3,1),(4,1),(5,2),(6,4) is OEIS A074021.
Column (1,5),(2,6),(3,3),(4,3),(5,6),(6,5) is OEIS A074022.
Column (1,6),(2,3),(3,5),(4,5),(5,3),(6,6) is OEIS A074023.