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Let $S$ be the antipode of a Hopf algebra $H$. The gauge invariance of Frobenius-Schur indicators implies that the traces of $S$ and $S^2$ are invariants of the finite tensor category Rep($H$). However, the question of whether the traces of all the powers of $S$ are gauge invariants is generally open. An affirmative answer of this question immediately implies the invariance of the order of $S$. In this talk, we will show that they are invariants of Rep($H$) when the Jacobson radical of $H$ is a Hopf ideal. This is a joint work with Cris Negron.