J_5, J_6$ restrict to Igusa’s generators [@Igusa], $\psi_4, \psi_6, \chi_{10}, \chi_{12}$ (and $J_4$ vanishes in this situation) [@CMSE7].

The periods of $H\oplus E_7\oplus E_7$ lattice polarized K3 surfaces $S$ in $H^2(S, \Z)$ determine points in ${\bf H}_2$. The Weierstrass coefficients of such elliptically fibered K3 surfaces were given in terms of $\Gamma_{\cal T}$-modular forms of even characteristic [@CMSE7].

The Weierstrass equation of a K3 surface with $H\oplus E_7\oplus E_7$ lattice polarization is given by [@CMSE7]: $$\label{Weierstrass with param in 2.2}
y^2 = x^3+ (e\, t^4+c\, t^3+a\, t^2) x + t^7+g\, t^6+(d\, e+f)\, t^5+c\, d\, t^4 + b\, t^3.$$ Up to some scale factors, the coefficients are given in terms of the modular forms $J_2, J_3, J_4, J_5, J_6$ [@CMSE7]: $$\begin{aligned}
\label{coeffs in 2.2}
c= -J_5(\omega), & d=-\frac{1}{3}J_4(\omega), & e = -3J_2(\omega) \\ \nonumber
f= J_6(\omega), & &  g=-2J_3(\omega) \\ \nonumber 
a=-3 d^2=-\frac{1}{3}J_4(\omega)^2, & & b=-2d^3=\frac{2}{27}J_4(\omega)^3.\end{aligned}$$ The elliptically fibered K3 surface determines a point in $D_{2,4}$, and this also determines a point in ${\bf H}_2$ under the isomorphism (\[H2 isom in 2.2\]), which we denote by $\omega$.

Now, consider a manifold $M$ and a line bundle $\Lambda$ on $M$, and choose sections $a,b,c,d,e,f, g$ of the line bundles $\Lambda^{\otimes 16}$, $\Lambda^{\otimes 