I will talk about recent results on sphere packing and energy minimization problems in dimensions $8$ and $24$. In these results a key role is played by a particular interpolation formula that allows one to reconstruct any radial Schwartz function from discrete samples of it and its Fourier transform. Surprisingly, this formula and its proof are closely connected to classical modular forms. I will explain this connection and also overview some of the more recent results on the closely related concept of Fourier uniqueness pairs.