Cartesian velocity moments of the Boltzmann equation are evaluated using modal solutions to the spherical harmonic oscillator as a basis set. The nonlinear collision matrix describing the interaction between any two modes is evaluated analytically for the Landau collision operator, and matrix elements describing collisions between identical particles are calculated for some pairs of azimuthally symmetric modes. First‐order linear transport coefficients calculated directly from collision matrix elements are shown to agree exactly with previously published results; coefficients of thermal conductivity and viscosity are computed much more accurately by trivially extending this calculation. Relaxation times for self‐collisions in a two‐dimensional linearized plasma are also computed, indicating that the plasma equilibrates in roughly one to ten times the Spitzer self‐collision time. The results obtained in this paper are useful for both analytic and numerical simulations of nonequilibrium plasmas and an explicit six‐moment model for a one‐component azimuthally symmetric plasma is given.