The cohomology of the moduli space M_{g,n} of smooth n-pointed complex curves of genus g carries mixed Hodge structures. Their top weight part has attracted attention due to its combinatorial nature, which connects it to the topology of its tropicalization, as shown by Chan, Galatius and Payne. In this talk, I would like to present an alternative approach to the study of the top weight cohomology in the case g=2 using local systems. This is joint work in progress with Dan Petersen.