General questions on the eigenfunctions of Laplacian and Dirac operators

Question

We know that the eigenvalues of the Laplacian contains a lot of information of a Riemannian manifold, but they do not determine the full information ( Hearing the shape of a drum). And the eigenfunctions of the Laplacian seem to have much more information (see the reference). Now my question is that whether the eigenfunctions of the Dirac operator would contain more information than that of Laplacian, since it seems to me that the Dirac operator is a more refined version of the Laplacian ( a Riemmanian manifold could have several spin structure). Here the Laplacian means the Laplace-Beltrami operator and the Dirac operator means the Dirac operator on the spinor bundle. Thank you.