  For any positive integers $n \ge 3$ and $r \ge 1$, we prove that the number
of monic irreducible polynomials of degree $n$ over $\mathbb{F}_{2^r}$ in which
the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has
period $24$ as a function of $n$, after a suitable normalization. A similar
result holds over $\mathbb{F}_{5^r}$, with the period being $60$. We also show
that this is a phenomena unique to characteristics $2$ and $5$. The result is
strongly related to the supersingularity of certain curves associated with
cyclotomic function fields, and in particular it complements an
equidistribution result of Katz.
