  We show that the eighth power of the Jacobi triple product is a
Jacobi--Eisenstein series of weight $4$ and index $4$ and we calculate its
Fourier coefficients. As applications we obtain explicit formulas for the
eighth powers of theta-constants of arbitrary order and the Fourier
coefficients of the Ramanujan Delta-function $\Delta(\tau)=\eta^{24}(\tau)$,
$\eta^{12}(\tau)$ and $\eta^{8}(\tau)$ in terms of Cohen's numbers $H(3,N)$ and
$H(5,N)$. We give new formulas for the number of representations of integers as
sums of eight higher figurate numbers. We also calculate the sixteenth and the
twenty-fourth powers of the Jacobi theta-series using the basic Jacobi forms.
