  We extend the work of Hellerman (arxiv:0902.2790) to derive an upper bound on
the conformal dimension $\Delta_2$ of the next-to-lowest nontrival primary
operator in unitary two-dimensional conformal field theories without chiral
primary operators. The bound we find is of the same form as found for
$\Delta_1$: $\Delta_2 \leq c_{tot}/12 + O(1)$. We find a similar bound on the
conformal dimension $\Delta_3$, and present a method for deriving bounds on
$\Delta_n$ for any $n$, under slightly modified assumptions. For asymptotically
large $c_{tot}$ and fixed $n$, we show that $\Delta_n \leq
\frac{c_{tot}}{12}+O(1)$. We conclude with a brief discussion of the
gravitational implications of these results.
