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@user43326: I think so: $c_1(\rho)\in H^2(BZ_{2r};\mathbb{Z})=\mathbb{Z}_{2r}$ represents 1, so the mod 2 reduction (which is the 2nd Stiefel-Whitney class of the real 2-dimensional bundle) represents 1 in $H^2(BZ_{2r};\mathbb{F}_2)=\mathbb{F}_2$ as well.
