A procedure to accurately simulate a free hot supersonic jet and its associated noise, which uses simultaneously a turbulence tripping method and a two-way coupling between a flow solver and a nonlinear acoustic solver, is proposed in this study. A Mach 3.1 overexpanded hot jet is computed via a large-eddy simulation by solving the filtered Navier-Stokes equations with a finite volume method on unstructured grids. The resulting noise is propagated in the far field by solving the full Euler equations with a high-order discontinuous Galerkin methodon unstructured grids. The full convergent-divergent nozzle is explicitly included in the computational domain thanks to the unstructured flow solver. Both a refined grid and a geometrical boundary layer tripping in the convergent are used to get highly disturbed turbulent conditions at the nozzle lips. The flow field appears to agree with the expected turbulence behavior and the available experimental data. The jet development shows significant improvement compared to similar past simulations. The far field acoustic levels are finely recovered at most of observation angles. An analysis of the acoustic near and far fields is then performed. The studied conditions lead to strong shock-associated noise and Mach wave emission. The spatio-frequency and azimuthal content of the acoustic field is described in order to identify the main noise properties. A particular noise component, different from screech tones and radiating upstream like Mach waves, is highlighted. Nonlinear propagation effects are finally quantified through specific metrics. They are found significant in both the near and the far fields which justifies the use of a nonlinear acoustic solver.