Spherical designs and modular forms of the D_4 lattice

In this paper, we study shells of the D_4 lattice with a slight generalization of spherical t -designs due to Delsarte–Goethals–Seidel, namely, the spherical design of harmonic index T (spherical T -design for short) introduced by Delsarte-Seidel. We first observe that, for any positive integer m , the 2 m -shell of D_4 is an antipodal spherical {10,4,2} -design on the three dimensional sphere. We then prove that the 2-shell, which is the D_4 root system, is a tight {10,4,2} -design, using the linear programming method. The uniqueness of the D_4 root system as an antipodal spherical {10,4,2} -design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the D_4 lattice in terms of orthogonal transformations of the D_4 root system, and the uniqueness of the D_4 lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the D_4 lattice and non-vanishing of the Fourier coefficients of a certain newform of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed.