In this paper, the theory of $\epsilon$-constants associated to tame finite group actions on arithmetic surfaces is used to define a Brauer group invariant $\mu(\X,G,V)$ associated to certain symplectic motives of weight one. The relationship between this invariant and $w_2(\pi)$ (the Galois-theoretic invariant associated to tame covers of surfaces defined by Cassou-Noguès, Erez and Taylor) is also discussed.