The twisted $L^2$-Euler characteristic is a homotopy invariant of CW complexes introduced in a 2018 article by Friedl and Lück. Since the invariant agrees with the Thurston norm on a large class of 3-manifolds, it appears quite promising for the study of fibrations over the circle in more general spaces, especially higher dimensional manifolds. We present an algorithm that computes the twisted $L^2$-Euler characteristic, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonné determinant of certain matrices. The algorithm needs to run for an extremely long time to certify its outputs, but a truncated, human-assisted version produces very good results in many cases, including hyperbolic link complements, closed census 3-manifolds, free-by-cyclic groups, and higher-dimensional examples, such as the fiber of the Ratcliffe-Tschantz 5-manifold.