  Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs
call for calculation of the expectation values of local operators in highly
excited energy eigenstates. This can be done efficiently by representing zero
modes of these operators in terms of the Virasoro algebra generators. In this
paper we present a pedagogical introduction explaining how this calculation can
be performed analytically or using computer algebra. We illustrate the
computation of zero modes by a number of examples and list explicit expressions
for all local operators from the vacuum family with the dimension of less or
equal than eight. Finally, we derive an explicit expression for the quantum KdV
generator $Q_7$ in terms of the Virasoro algebra generators. The obtained
results can be used for quantitative studies of ETH at finite value of central
charge.
