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We give an explicit conjectural formula for the motivic Euler characteristicof an arbitrary symplectic local system on the moduli space A_3 of principallypolarized abelian threefolds. The main term of the formula is a conjecturalmotive of Siegel modular forms of a certain type; the remaining terms admit asurprisingly simple description in terms of the motivic Euler characteristicsfor lower genera. The conjecture is based on extensive counts of curves ofgenus three and abelian threefolds over finite fields. It provides a lot of newinformation about vector-valued Siegel modular forms of degree three, such asdimension formulas and traces of Hecke operators. We also use it to predictseveral lifts from genus 1 to genus 3, as well as lifts from G_2 and newcongruences of Harder type.

Siegel modular forms of degree three and the cohomology of local systems