The Hardy space H²(D²) can be viewed as a module over the polynomial ring C[z,w] with module action defined by multiplication of functions. The core operator is a bounded self-adjoint integral operator defined on submodules of H²(D²), and it gives rise to some interesting numerical invariants for the submodules. These invariants are difficult to compute or estimate in general. This paper computes these invariants for homogeneous submodules through Toeplitz determinants.