  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 $2m$-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.

Combinatorics;Number Theory