  For the hypoelliptic differential operators $P={\partial^2_ x}+(x^k\partial_
y -x^l{\partial_t})^2$ introduced by T. Hoshiro, generalizing a class of M.
Christ, in the cases of $k$ and $l$ left open in the analysis, the operators
$P$ also fail to be {\em{analytic}} hypoelliptic (except for $(k,l)=(0,1)$), in
accordance with Treves' conjecture. The proof is constructive, suitable for
generalization, and relies on evaluating a family of eigenvalues of a
non-self-adjoint operator.
