  The modified elliptic genus for an M5-brane wrapped on a four-cycle of a
Calabi-Yau threefold encodes the degeneracies of an infinite set of BPS states
in four dimensions. By holomorphy and modular invariance, it can be determined
completely from the knowledge of a finite set of such BPS states. We show the
feasibility of such a computation and determine the exact modified elliptic
genus for an M5-brane wrapping a hyperplane section of the quintic threefold.
