  We obtain new restrictions on the linear programming bound for sphere
packing, by optimizing over spaces of modular forms to produce feasible points
in the dual linear program. In contrast to the situation in dimensions 8 and
24, where the linear programming bound is sharp, we show that it comes nowhere
near the best packing densities known in dimensions 12, 16, 20, 28, and 32.
More generally, we provide a systematic technique for proving separations of
this sort.
