  The well known conformal covariance of the Dirac operator acting on spinor
fields over a semi Riemannian spin manifold does not extend to powers thereof
in general. For odd powers one has to add lower order curvature correction
terms in order to obtain conformal covariance. We derive an algorithmic
construction in terms of associated tractor bundles to compute these correction
terms. Depending on the signature of the semi Riemannian manifold in question,
the obtained conformal powers of the Dirac operator turn out to be formally
self-adjoint with respect to the $L^2-$scalar product, or formally
anti-self-adjoint, respectively. Working out this algorithm we present explicit
formulas for the conformal third and fifth power of the Dirac operator.
  Furthermore, we present a new family of conformally covariant differential
operators acting on the spin tractor bundle which are induced by conformally
covariant differential operators acting on the spinor bundle.
  Finally, we will give polynomial structures for the first examples of
conformal powers in terms of first order differential operators acting on the
spinor bundle.
