Sample Problems

Let $n$ beads be arranged on a circle and indexed $0,1,\dots,n-1$. Each bead is colored by an element of the finite field $\mathbb{F}_7=\mathbb{Z}/7\mathbb{Z}$. Two colorings $(c_0,\dots,c_{n-1})\in \mathbb{F}_7^n$ are considered equivalent if one can be obtained from the other by a rotation or reflection of the necklace (that is, by the usual dihedral action $D_n$ on indices).

A coloring is admissible if it satisfies all three linear constraints in $\mathbb{F}_7$:

For this problem, take $n=30$. How many $D_n$-equivalence classes of admissible colorings are there for $n=30$? Report your answer modulo $1{,}000{,}003$ as a positive integer.