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We show how to calculate the Euler characteristic of a local systemV  λassociated to an irreducible representationV  λof the symplectic group of genus 3 on the moduli spaceM  3of curves of genus 3 and the moduli spaceA  3of principally polarised abelian varieties of dimension 3.

The Euler characteristic of local systems on the moduli of curves and abelian varieties of genus three