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A bstract We consider excited states in a CFT, obtained by applying a weak unitary perturbation to the vacuum. The perturbation is generated by the integral of a local operator J ( n ) of modular weight n over a spacelike surface passing through x = 0. For | n | ≥ 2 the modular Hamiltonian associated with a division of space at x = 0 picks up an endpoint contribution, sensitive to the details of the perturbation (including the shape of the spacelike surface) at x = 0. The endpoint contribution is a sum of light-ray moments of the perturbing operator J ( n ) and its descendants. For perturbations on null planes only moments of J ( n ) itself contribute.