  Real space renormalization group maps, e.g., the majority rule
transformation, map Ising type models to Ising type models on a coarser
lattice. We show that each coefficient of the renormalized Hamiltonian in the
lattice gas variables depends on only a finite number of values of the
renormalized Hamiltonian. We introduce a method which computes the values of
the renormalized Hamiltonian with high accuracy and so computes the
coefficients in the lattice gas variables with high accuracy. For the critical
nearest neighbor Ising model on the square lattice with the majority rule
transformation, we compute over 1,000 different coefficients in the lattice gas
variable representation of the renormalized Hamiltonian and study the decay of
these coefficients. We find that they decay exponentially in some sense but
with a slow decay rate. We also show that the coefficients in the spin
variables are sensitive to the truncation method used to compute them.
