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

Companion forms and explicit computation of PGL2 number fields with very little ramification