The Divisibility of $\mathrm{GL}(n, q)$ Character Values

Let $q$ be a prime power, and $d$ a positive integer. We study the proportion of irreducible characters of $\mathrm{GL}(n,q)$ whose values evaluated on a fixed matrix $g$ are divisible by $d$. As $n$ approaches infinity, this proportion tends to $1$ when $q$ is coprime to $d$. When $q$ and $d$ are not coprime, and $g=1$, this proportion is bounded above by $\frac{1}{q}$.