  The aim of the present paper is to give evidence, largely numerical, in
support of the non-commutative main conjecture of Iwasawa theory for the motive
of a primitive modular form of weight k>2 over the Galois extension of Q
obtained by adjoining to Q all p-power roots of unity, and all p-power roots of
a fixed integer m>1. The predictions of the main conjecture are rather
intricate in this case because there is more than one critical point, and also
there is no canonical choice of periods. Nevertheless, our numerical data
agrees perfectly with all aspects of the main conjecture, including Kato's
mysterious congruence between the cyclotomic Manin p-adic L-function, and the
cyclotomic p-adic L-function of a twist of the motive by a certain non-abelian
Artin character of the Galois group of this extension.
