We investigate the influence of eccentricity on linear stability of purely elastic Dean flow of an Upper Convected Maxwell liquid. A pseudo-spectral Chebyshev-Fourier collocation (CFC) technique, that exploits smoothness of the computational domain, periodicity in the azimuthal direction and exponential convergence characteristics of spectral approximations, is employed for the spatial discretization of the governing equations. Arnoldi subspace iteration technique is employed for the selective evaluation of the leading eigenvalues. The CFC method was first benchmarked successfully for two limiting cases that correspond to Dean flow and plane Poiseuille flow. The eigenspectrum of Dean flow is shown to consist of a number of spatially and temporally near-resonant modes with critical Deborah numbers close to each other, the axisymmetric and stationary eigenmode being the most dangerous, in agreement with earlier analysis [6]. Results obtained for eccentric Dean flow for relatively small gap width show that eccentricity, epsilon, has a non-monotonic influence on the Linear stability of Dean flow. The critical Deborah number first increases with increasing epsilon for epsilon less than or equal to 0.1 and decreases with increasing epsilon for epsilon>0.1. The critical eigenfunctions are three-dimensional and stationary with a very high degree of spacial non-uniformity. They manifest as three-dimensional 'rolls' packed closely along the circumference of the cylinders. These complex structures exhibit steep streamwise and radial gradients near the wall and in the bulk, necessitating fine spatial resolution in the computations. Potential mechanisms of instability are discussed.