A statistical equilibrium theory in two-dimensional turbulence is used to study the emergence of coherent structures. Macroscopic states are described by a local probability measure on the fluctuating vorticity field. The most probable macroscopic state is characterized by maximizing entropy subject to a family of constraints derived from the conserved quantities of the incompressible Euler equations. Coherent structures are identified with such macrostates. Attention is focused on the special case of enstrophy-moments and a doubly periodic domain. The algorithm of Turkington and Whitaker is applied to this special case. The convergence properties of the algorithm are derived from the optimization structure of the constraint maximization problem. The algorithm produces an entropy increasing sequence. The trivial case of conserving energy and enstrophy serves as the starting equilibrium state when implementing the algorithm. The probability density of the energy-enstrophy model is Gaussian and we perturb away from the trivial case by imposing higher enstrophy moments. Perturbing away from the trivial case forces the model to become nonlinear. It is found that the preferred statistical equilibrium state of the maximum entropy problem is determined by kurtosis. For high kurtosis a dipole is preferred and for low kurtosis a shear-layer is preferred. Several local solutions are found which converge to the maximum entropy state for high energy.