In 2016, Viazovska proved that the E₈ lattice gives the densest sphere packing problem in 8-dimesions [V16]. Shortly thereafter, Cohn, Kumar, Miller, Radchenko, and Viazovska [CKM+16] proved that the Leech lattice gives the densest sphere packing in 24-dimensions. Their proofs find feasible solutions to the the Cohn-Elkies LP-method [CE03] to give upper bounds on the density of sphere packings in 8 and 24 dimensions that match sphere packings centered on the E8 and Leech lattice respectively. We prove that in contrast to the situation in dimensions 8 and 24, the LP-method is insufficient to prove that the densest known sphere packing is indeed the densest sphere packing in 12, 16, 20, 28, and 32 dimensions. In our approach, the obstructions comes from modular forms whose q-expansions have non-negative coefficients, including many coefficients equal to zero at low indices. We describe a computational approach for lower-bounding the power of the LP-method that involves linear programming over spaces of modular forms and tricks for producing modular forms with coefficients exactly equal to zero from forms with coefficients which are only approximately equal to zero.