The behaviour of an isolated polymer floating in a solvent forms the basis of our understanding of polymer dynamics(1,2). Classical theories describe the motion of a polymer with linear equations of motion, which yield a set of ’normal modes’, analogous to the fundamental frequency and the harmonics of a vibrating violin string. But hydrodynamic interactions make polymer dynamics inherently nonlinear, and the linearizing approximations required for the normal-mode picture have therefore been questioned(1). Here we test the normal-mode theory by measuring the fluctuations of single molecules of DNA held in a partially extended state with optical tweezers. We find that the motion of the DNA can be described by linearly independent normal modes, and we have experimentally determined the eigenstates of the system. Furthermore, we show that the spectrum of relaxation times obeys a power law.