  We give an explicit description of the matrix associated to the $U_p$
operator acting on spaces of overconvergent Hilbert modular forms over totally
real fields. Using this, we compute slopes for weights in the centre and near
the boundary of weight space for certain real quadratic fields.
\added[id=h]{Near the boundary of weight space we see that the slopes do not
appear to be given by finite unions of arithmetic progressions but instead can
be produced by a simple recipe from which we make a conjecture on the structure
of slopes. We also prove a lower bound on the Newton polygon of the $U_p$.
