The harmonic strength of shells of lattices and weighted theta series

This is the write-up of a talk given in RIMS conference ``Analytic and arithmetic aspects of automorphic representations", where I outlined two kinds of different results related to the D4 lattice, obtained in a joint work with Hirao and Nozaki. In general, from the design theoretical viewpoints, we are interested in determining the harmonic strength of finite subsets on the unit sphere. When considering the shells of a lattice, this problem shifts to the non-vanishing problem of the Fourier coefficients of elliptic cusp forms. Various instances, mostly from root lattices, are listed in this note.