I'm trying to calculate the $\Theta$ series of $E_8$ lattice, using the following Gram matrix (the Cartan Matrix of $E_8$): $$\left(\begin{matrix} 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 2 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0\\ -1 & 0 & -1 & 2 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 2 \end{matrix}\right)$$
Since now I'm dealing with more than two variables, it's not always possible to complete the squares (at least I couldn't succeed in this case), and write the theta series in the form of $\Theta(z)=\sum_{x_1,...,x_n}q^{x_1^2+...+x_n^2}$ ($q=e^{i\pi z}$).
What I got for the power of $q$ is
$$2x_1^2 + 2x_2^2 + 2x_3^2 + 2x_4^2 + 2x_5^2 + 2x_6^2 + 2x_7^2 + 2x_8^2- 2x_1x_4-2x_2x_3-2x_3x_4-2x_4x_5-2x_5x_6-2x_6x_7-2x_7x_8\\ = (x_2-x_3)^2 + (x_3-x_4)^2 + (x_4-x_5)^2 + (x_5-x_6)^2 + (x_6-x_7)^2 + (x_7-x_8)^2 + (x_1-x_4)^2 \\- x_4^2 + x_8^2$$
Although it seems that I managed to rewrite the power of $q$ in a sum of squares, they are not all independent from each other (8 variables before and 9 squares now). I have no idea how to finish the calculation and get the final result $\Theta_{E_8}(z)=\frac{1}{2}\left(\theta_2(z)^8+\theta_3(z)^8+\theta_4(z)^8\right)$.
An additional general question: how to calculate theta series of a lattice using Gram matrix, when I have many (>2) variables and can't simply complete the square (or diagonalise the Gram matrix)?
Any help specifically to this question or general suggestion is appreciated.