MSG 81.34

\[\begin{align*} &E_{2}^{1,-1}=\mathbb{Z}\\ &E_{1}^{0,-1}=\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{A}_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{A}_3}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{A}_4}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{B}_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{B}_2}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{C}_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{C}_3}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{C}_4}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{D}_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{D}_3}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{D}_4}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{E}_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{E}_2}]\oplus\mathbb{Z}[\boldsymbol{b}^{(0)}_{\text{F}_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{F}_3}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(0)}_{\text{F}_4}]\\ &E_{1}^{1,-1}=\mathbb{Z}_2[\boldsymbol{b}^{(1)}_{a_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(1)}_{b_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(1)}_{c_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(1)}_{d_1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(1)}_{e_2}]\oplus\mathbb{Z}[\boldsymbol{b}^{(1)}_{f_1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(1)}_{f_2}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(1)}_{g_2}]\\ &E_{1}^{2,-1}=\mathbb{Z}_2[\boldsymbol{b}^{(2)}_{\alpha _1}]\oplus\mathbb{Z}_2[\boldsymbol{b}^{(2)}_{\beta _1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(2)}_{\gamma _1}]\oplus\mathbb{Z}[\boldsymbol{b}^{(2)}_{\delta _1}] \end{align*}\]

\[\begin{align*} &[X^{(1)}]^{-1}=\left( \begin{array}{cccccccc} a_1 & b_1 & c_1 & d_1 & e_2 & f_1 & f_2 & g_2 \\ 1 & -1 & 0 & 0 & -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\ \end{array} \right)\\ &[V^{(0)}]^{-1}=\left( \begin{array}{cccccccccccccccccccccc} \text{A}_1 & \text{A}_3 & \text{A}_4 & \text{B}_1 & \text{B}_2 & \text{C}_1 & \text{C}_3 & \text{C}_4 & \text{D}_1 & \text{D}_3 & \text{D}_4 & \text{E}_1 & \text{E}_2 & \text{F}_1 & \text{F}_3 & \text{F}_4 & a_1 & b_1 & c_1 & d_1 & e_2 & g_2 \\ 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 & 1 & 1 & 2 & -2 & 0 & 0 & -2 & 2 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 2 & 0 & 0 & 0 & -2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & -1 & -1 & -1 & 0 & 0 & 2 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right)\\ &\Lambda^{(0)}=\left( \begin{array}{ccccccc} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\ &[V^{(1)}]^{-1}=\left( \begin{array}{cccccccccc} a_1 & b_1 & c_1 & d_1 & e_2 & f_1 & f_2 & g_2 & \alpha _1 & \beta _1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right)\\ &\Sigma^{(1)}=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right) \end{align*} \]