  We present an algorithm that computes Friedl and L\"uck's twisted $L^2$-Euler
characteristic for a suitable regular CW complex, employing Oki's matrix
expansion algorithm to indirectly evaluate the Dieudonn\'e determinant. 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,
such as hyperbolic link complements, closed census 3-manifolds, free-by-cyclic
groups, and higher-dimensional examples, such as the fiber of the
Ratcliffe-Tschantz manifold.
