Suspensions of swimming microorganisms are a class of active suspensions that show an interesting rheological response in steady shear flow. In particular, the particle contribution to the viscosity can be negative, which has been calculated from models and measured experimentally. In this article, the material functions in large-amplitude oscillatory shear (LAOS) flow are calculated. In addition to the linear material functions, the nonlinearities are quantified analytically using the intrinsic nonlinear material functions. The particle contribution to both the storage and loss modulus can be negative. Since the suspending fluid is assumed Newtonian (and so has no storage modulus), the overall storage modulus can be negative. The intrinsic nonlinearities also show differences between passive and active suspensions. At small frequency, the active swimming can change the sign of the material functions. However, the viscous material functions are independent of the swimming motion at a very large frequency. The changes in sign of the material functions and the unique dependence on frequency may act as a rheological fingerprint of suspensions of swimming organisms.