  Newman showed that for primes $p\geq 5$ an integral circulant determinant of
prime power order $p^t$ cannot take the value $p^{t+1}$ once $t\geq 2.$ We show
that many other values are also excluded. In particular, we show that $p^{2t}$
is the smallest power of $p$ attained for any $t\geq 3$, $p\geq 3.$ We
demonstrate the complexity involved by giving a complete description of the
$25\times 25$ and $27\times 27$ integral circulant determinants. The former
case involves a partition of the primes that are $1\bmod5$ into two sets,
Tanner's \textit{perissads} and \textit{artiads}, which were later
characterized by E. Lehmer.
