  We give new explicit formulas for the representations of the mapping class
group of a genus one surface with one boundary component which arise from
Integral TQFT. Our formulas allow one to compute the h-adic expansion of the
TQFT-matrix associated to a mapping class in a straightforward way. Truncating
the h-adic expansion gives an approximation of the representation by
representations into finite groups. As a special case, we study the induced
representations over finite fields and identify them up to isomorphism. The key
technical ingredient of the paper are new bases of the Integral TQFT modules
which are orthogonal with respect to the Hopf pairing. We construct these
orthogonal bases in arbitrary genus, and briefly describe some other
applications of them.
