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We had previously defined the rho invariant ρ_spin(Y,E,H, g) for the twisted Dirac operator ∂^E_H on a closed odd dimensional Riemannian spin manifold (Y, g), acting on sections of a flat hermitian vector bundle E over Y, where H = ∑ i^j+1 H_2j+1 is an odd-degree differential form on Y and H_2j+1 is a real-valued differential form of degree 2j+1. Here we show that it is a conformal invariant of the pair (H, g). In this paper we express the defect integer ρ_spin(Y,E,H, g) - ρ_spin(Y,E, g) indoi:10.4171/jncg/209 fatcat:zjvsu2c7qfdc7kk7uzkr63tltm