  We show that the eigenschemes of $4 \times 4 \times 4$ symmetric tensors are
parametrized by a linear subvariety of the Grassmannian
$\operatorname{Gr}(3,\mathbb{P}^{14})$. We also study the decomposition of the
eigenscheme into the subscheme associated to the zero eigenvalue and its
residue. In particular, we categorize the possible degrees and dimensions.
