  We introduce mod 3 triple Milnor invariants and triple cubic residue symbols
for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$,
following the analogies between knots and primes. Our triple symbol generalizes
both the cubic residue symbol and R\'{e}dei's triple symbol, and describes the
decomposition law of a prime in a mod 3 Heisenberg extension of degree 27 over
$\mathbb{Q}(\sqrt{-3})$ with restricted ramification, which we construct
concretely in the form similar to R\'{e}dei's dihedral extension over
$\mathbb{Q}$. We also give a cohomological interpretation of our symbols by
triple Massey products in Galois cohomology.
