  It has recently been conjectured that the elliptic genus of K3 can be written
in terms of dimensions of Mathieu group M24 representations. Some further
evidence for this idea was subsequently found by studying the twining genera
that are obtained from the elliptic genus upon replacing dimensions of Mathieu
group representations by their characters. In this paper we find explicit
formulae for all (remaining) twining genera by making an educated guess for
their general modular properties. This allows us to identify the decomposition
of all expansion coefficients in terms of dimensions of M24-representations.
For the first 500 coefficients we verify that the multiplicities with which
these representations appear are indeed all non-negative integers. This
represents very compelling evidence in favour of the conjecture.
