  In previous works, we described algorithms to compute the number field cut
out by the mod ell representation attached to a modular form of level N=1. In
this article, we explain how these algorithms can be generalised to forms of
higher level N.
  As an application, we compute the Galois representations attached to a few
forms which are supersingular or admit a companion mod ell with ell=13 (and
soon ell=41), and we obtain previously unknown number fields of degree ell+1
whose Galois closure has Galois group PGL(2,ell) and a root discriminant that
is so small that it beats records for such number fields.
  Finally, we give a formula to predict the discriminant of the fields obtained
by this method, and we use it to find other interesting examples, which are
unfortunately out of our computational reach.
