  We consider families of theories with large N=4 superconformal symmetry. We
define an index generalizing the elliptic genus of theories with N=2 symmetry.
In contrast to the N=2 case, the new index constrains part of the non-BPS
spectrum. Motivated by aspects of the AdS/CFT correspondence we study the index
in the examples of symmetric product theories. We give a physical
interpretation of the Hecke operators which appear in the expressions for
partition functions of such theories. Finally, we compute the index for a
nontrivial example of a symmetric product theory.
