  We study the emergence of two-dimensional conformal symmetry in critical
quantum spin chains on the finite circle. Our goal is to characterize the
conformal field theory (CFT) describing the universality class of the
corresponding quantum phase transition. As a means to this end, we propose and
demonstrate automated procedures which, using only the lattice Hamiltonian $H =
\sum_j h_j$ as an input, systematically identify the low-energy eigenstates
corresponding to Virasoro primary and quasiprimary operators, and assign the
remaining low-energy eigenstates to conformal towers. The energies and momenta
of the primary operator states are needed to determine the primary operator
scaling dimensions and conformal spins -- an essential part of the conformal
data that specifies the CFT. Our techniques use the action, on the low-energy
eigenstates of $H$, of the Fourier modes $H_n$ of the Hamiltonian density
$h_j$. The $H_n$ were introduced as lattice representations of the Virasoro
generators by Koo and Saleur [Nucl. Phys. B 426, 459 (1994)]. In this paper we
demonstrate that these operators can be used to extract conformal data in a
nonintegrable quantum spin chain.
