Flow coefficients nu(n) for n = 2, 3, 4, characterizing the anisotropic collective flow in Au + Au collisions at root s(NN) = 200 GeV, are measured relative to event planes Psi(n), determined at large rapidity. We report nu(n) as a function of transverse momentum and collision centrality, and study the correlations among the event planes of different order n. The nu(n) are well described by hydrodynamic models which employ a Glauber Monte Carlo initial state geometry with fluctuations, providing additional constraining power on the interplay between initial conditions and the effects of viscosity as the system evolves. This new constraint can serve to improve the precision of the extracted shear viscosity to entropy density ratio eta/s.