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Despite the apparent complexity of turbulent flow, identifying a simpler description of the underlying dynamical system remains a fundamental challenge. Capturing how the turbulent flow meanders amongst unstable states (simple invariant solutions) in phase space, as envisaged by Hopf in 1948, using some efficient representation offers the best hope of doing this, despite the inherent difficulty in identifying these states. Here, we make a significant step towards this goal by demonstrating that deep convolutional autoencoders can identify low-dimensional representations of two-dimensional turbulence which are closely associated with the simple invariant solutions characterizing the turbulent attractor. To establish this, we develop latent Fourier analysis that decomposes the flow embedding into a set of orthogonal latent Fourier modes which decode into physically meaningful patterns resembling simple invariant solutions. The utility of this approach is highlighted by analysing turbulent Kolmogorov flow (flow on a 2D torus forced at large scale) at $Re=40$ where, in between intermittent bursts, the flow resides in the neighbourhood of an unstable state and is very low dimensional. Projections onto individual latent Fourier wavenumbers reveal the simple invariant solutions organising both the quiescent and bursting dynamics in a systematic way inaccessible to previous approaches.