[en] We extend the work of Hellerman http://dx.doi.org/10.1007/JHEP08(2011)130 to derive an upper bound on the conformal dimension Δ_2 of the next-to-lowest nontrival primary operator in unitary, modular-invariant two-dimensional conformal field theories without chiral primary operators, with total central charge c_t_o_t>2. The bound we find is of the same form as found by Hellerman for Δ_1: Δ_2≤((c_t_o_t)/12)+O(1). We obtain a similar bound on the conformal dimension Δ_3, and present a method for deriving bounds on Δ_n for any n, under slightly modified assumptions. For asymptotically large c_t_o_t and n≲exp (πc/12), we show that Δ_n≤((c_t_o_t)/12)+O(1). This implies an asymptotic lower bound of order exp (πc_t_o_t/12) on the number of primary operators of dimension ≤c_t_o_t/12+O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G_N