  Given a critical quantum spin chain described by a conformal field theory
(CFT) at long distances, it is crucial to understand the universal conformal
data. One most important ingredient is the operator product expansion (OPE)
coefficients, which describe how operators fuse into each other. It has been
proposed in [Zou, Vidal, Phys. Rev. B 105, 125125] that the OPE coefficients
can be computed from overlaps of low-energy wavefunctions of the spin chain. In
this work, we establish that all conformal data including central charge,
conformal dimensions, and OPE coefficients are encoded in the wavefunction
overlaps, with universal finite-size corrections that depend on the operator
content of the cyclic orbifold CFT. Thus this method allows us to numerically
compute all the conformal data based solely on the low-energy eigenstates. The
predictions are verified in the Ising and XXZ model. As an application, we
study the recently proposed Haagerup model built from the Haagerup fusion
category. We find that the CFT has central charge $c \approx 2.1$ and the
lowest spin-$1$ operator in the twisted sector has scaling dimension $1 <
\Delta_J \leq 1.4$.
