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In this paper, we study shells of the $D_4$ lattice with a slightly general
concept 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 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 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 coefficient of a certain newforms of level 2.
Motivated by this, congruence relations for the Fourier coefficients are
discussed.