  Let p be any prime and a be a positive integer. For nonnegative integers l,n
and an integer r, the normalized cyclotomic $\psi$-coefficient
$${n,r}_{l,p^a}:=p^{-[(n-p^{a-1}-lp^a)/(p^{a-1}(p-1))]} \sum_{k=r(mod
p^a)}(-1)^k{n \choose k}{{(k-r)/p^a} \choose l}$$ is known to be an integer. In
this paper, we show that this coefficient behaves like binomial coefficients
and satisfies some Lucas-type congruences. This implies that a congruence of
Wan is often optimal, and two conjectures of Sun and Davis are true.
