The cohomology of arithmetic groups is made up of two pieces, the cuspidal and noncuspidal parts. Within the cuspidal cohomology is a subspace— the f-cuspidal cohomology—spanned by the classes that generate representations of the associated finite Lie group which are cuspidal in the sense of finite Lie group theory. Few concrete examples of f-cuspidal cohomology have been computed geometrically, outside the cases of rational rank 1, or where the symmetric space has a Hermitian structure. This paper presents new computations of the f-cuspidal cohomology of principal congruence subgroups Γ ( p ) \Gamma (p) of GL ( 3 , Z ) {\text {GL}}(3,\mathbb {Z}) of prime level p. We show that the f-cuspidal cohomology of Γ ( p ) \Gamma (p) vanishes for all p ⩽ 19 p \leqslant 19 with p ≠ 11 p \ne 11 , but that it is nonzero for p = 11 p = 11 . We give a precise description of the f-cuspidal cohomology for Γ ( 11 ) \Gamma (11) in terms of the f-cuspidal representations of the finite Lie group GL ( 3 , Z / 11 ) {\text {GL}}(3,\mathbb {Z}/11) . We obtained the result, ultimately, by proving that a certain large complex matrix M is rank-deficient. Computation with the SVD algorithm gave strong evidence that M was rank-deficient; but to prove it, we mixed ideas from numerical analysis with exact computation in algebraic number fields and finite fields.