Let $ M \subset \mathbb{R}^2 $ be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and 120). What is the spectrum of the laplacian $ \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} $ with dirichlet boundary conditions on $ M $?

The spectrum of the laplacian on the equilateral triangle is known, so some of the eigenfunctions - those that vanish on the diagonal - are known. But what about the whole spectrum?