  We show that the Seifert-Weber dodecahedral space $\mathsf{SW}$ is an
$L$-space. The proof builds on our work relating Floer homology and spectral
geometry of hyperbolic three-manifolds. A direct application of our previous
techniques runs into difficulties arising from the computational complexity of
the problem. We overcome this by exploiting the large symmetry group and the
arithmetic and tetrahedral group structure of $\mathsf{SW}$ to prove that small
eigenvalues on coexact $1$-forms must have large multiplicity.
