Concepts from quantum information theory have become increasingly important in our understanding of entanglement in QFTs. One prominent example of this is the modular Hamiltonian, which is closely related to the Unruh effect. Using complex analysis, we determine this operator for the chiral fermion at finite temperature on the circle and show that it exhibits surprising new features. This simple system illustrates how a modular flow can transition from complete locality to complete non-locality, thus bridging the gap between previously known limits. We derive the first exact results for the entropy in the different spin sectors, and comment on the analytic continuation of the Rényi entropies to the complex plane.