Let D be an equilateral triangle of side 1. We consider solutions of $\Delta u + \lambda u = 0$ in D with either the boundary condition $u = 0$ or ${{\partial u} / {\partial n }} = 0$. Let $n(\lambda )$ be the number of distinct eigenvalues $ \leqq \lambda $, $N(\lambda )$ be the total number of eigenvalues $ \leqq \lambda $, including multiplicities. Theorem 1 states that for either boundary condition, $\lambda _{mn} = ({{16\pi ^2 } / {27}})(m^2 + n^2 - mn)$, where $m + n \equiv 0(\bmod 3)$. In the first case it is further required that $m \ne 2n$. Theorem 2 states that $\lim _{\lambda \to \infty } ({{N(\lambda )} / {n(\lambda )}}) = \infty $. The proof uses the representation of $\lambda _{mn} $ as the norm of an integer in the quadratic number field $k(\omega )$, where $\omega $ is a primitive cube root of unity. These results contrast with the generic results for domains with $Z_3$ symmetry obtained by V. Arnold (Functional Anal. Appl., 1972).