  We study Majorana zero modes bound to giant vortices in topological
superconductors or topological insulator/normal superconductor
heterostructures. By expanding in inverse powers of a large winding number $n$,
we find an analytic solution for asymptotically all $n$ zero modes required by
the index theorem. Contrary to the existing estimates, the solution is not
pinned to the vortex boundary and is composed of the warped lowest Landau level
states. While the dynamics which shapes the zero modes is a subtle interference
of the magnetic effects and Andreev reflection, the solution is very robust and
is determined by a single parameter, the vortex radius. The resulting local
density of states has a number of features which give remarkable signatures for
an experimental observation of the Majorana fermions in two dimensions.
