  We present new obstructions for a knot K in S^3 to admit purely cosmetic
surgeries, which arise from the study of Witten-Reshetikhin-Turaev invariants
at fixed level. In particular, we strengthen a recent result of Hanselman,
showing that if K has purely cosmetic surgeries then the slopes of the
surgeries are of the form 1/5k except if the Jones polynomial of K evaluated at
a 5-th root of unity is 1. For any odd prime r, we also give an obstruction for
K to have a 1/k surgery slope with k coprime to r that involves the values of
the first (r-3)/2 colored Jones polynomials of K at an r-th root of unity. We
verify the purely cosmetic surgery conjecture for all knots with at most 17
crossings.
