We study the linear eigenvalue problem for the distribution function associated with Hookean and FENE dumbbell models. For Hookean dumbbells, the eigenfunctions can be expressed by generalized Laguerre polynomials. The eigenvalue problem for the FENE dumbbell leads to a confluent Heun equation. The first few eigenvalues are calculated numerically. We also calculate these eigenvalues using perturbation of the Hookean case. We show how the knowledge of the eigenvalues and eigenfunctions can be used to construct the stress relaxation modulus.