The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

For a partition $lambda=\{lambda_1 \geq \lambda_2 \geq \lambda_3 \}$ of non-negative integers, we calculate the Euler characteristic of the local system $V_{\lambda}$ on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some $\lambda$ of low degree, we make a guess for the motivic Euler characteristic of $V_{\lambda}$ using counting of curves over finite fields.