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
8
and
24
24
, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions
12
12
,
16
16
,
20
20
,
28
28
, and
32
32
. More generally, we provide a systematic technique for proving separations of this sort.