Irreducible polynomials over ${\mathbb{F}}_{2^r}$ with three prescribed coefficients

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.