Eisenstein series twisted with non-expanding cusp monodromies

Let $\mathrm{\Gamma }$ be a geometrically finite Fuchsian group and suppose that $\chi \text{\hspace{0.17em}⁣}:\mathrm{\Gamma }\to \mathrm{G}\mathrm{L}(V)$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $\mathrm{\Gamma }$ with twist $\chi$ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.