Eisenstein series twisted with non-expanding cusp monodromies

Let $Gamma$ be a geometrically finite Fuchsian group and suppose that $chicolonGammatomathrm{GL}(V)$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $Gamma$ with twist $chi$ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.