Number of conjugacy classes in general affine group of fixed degree over a finite field is polynomial function of field size

Statement

Suppose $n$ is a natural number. Then, there exists a polynomial function $f$ of degree $n$ such that, for any prime power $q$ , the number of conjugacy classes in the general affine group $G A (n, q)$ (i.e., the general affine group of degree $n$ over the finite field of size $q$ ) is $f (q)$ .

Below, we list some general observations about the polynomial in $q$ giving number of conjugacy classes in $G A (n, q)$ .

Item	Value
Degree of polynomial	$n$
Leading coefficient of polynomial	1, i.e., it is always a monic polynomial
Factors of polynomial	no common factors to all polynomials
Coefficients of polynomial	The polynomial is an integer-valued polynomial, i.e., it sends integers to integers. Is it also true that all coefficients of the polynomial are integers? Seems so from first few examples

$n$ (degree of general affine group, degree of polynomial)	polynomial of $q$ giving number of conjugacy classes in $G A (n, q)$	More information
1	$q$	element structure of general affine group of degree one over a finite field
2	$q^{2} + q - 1$	element structure of general affine group of degree two over a finite field
3	$q^{3} + q^{2} - 1$	element structure of general affine group of degree three over a finite field
4	$q^{4} + q^{3} + q^{2} - q - 1$	element structure of general affine group of degree four over a finite field
5	?	element structure of general affine group of degree five over a finite field
6	?	element structure of general affine group of degree six over a finite field
7	?