  In 1947, Lehmer conjectured that the Ramanujan $\tau$-function $\tau (m)$
never vanishes for all positive integers $m$, where the $\tau (m)$ are the
Fourier coefficients of the cusp form $\Delta_{24}$ of weight 12. Lehmer
verified the conjecture in 1947 for $m<214928639999$. In 1973, Serre verified
up to $m<10^{15}$, and in 1999, Jordan and Kelly for $m<22689242781695999$.
  The theory of spherical $t$-design, and in particular those which are the
shells of Euclidean lattices, is closely related to the theory of modular
forms, as first shown by Venkov in 1984. In particular, Ramanujan's
$\tau$-function gives the coefficients of a weighted theta series of the
$E_{8}$-lattice. It is shown, by Venkov, de la Harpe, and Pache, that $\tau
(m)=0$ is equivalent to the fact that the shell of norm $2m$ of the
$E_{8}$-lattice is an 8-design. So, Lehmer's conjecture is reformulated in
terms of spherical $t$-design.
  Lehmer's conjecture is difficult to prove, and still remains open. In this
paper, we consider toy models of Lehmer's conjecture. Namely, we show that the
$m$-th Fourier coefficient of the weighted theta series of the
$\mathbb{Z}^2$-lattice and the $A_{2}$-lattice does not vanish, when the shell
of norm $m$ of those lattices is not the empty set. In other words, the
spherical 5 (resp. 7)-design does not exist among the shells in the
$\mathbb{Z}^2$-lattice (resp. $A_{2}$-lattice).
