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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.
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