### M Theory Lesson 130

On page 149, Ebeling mentions that the characteristic polynomial (weight enumerator) of a self-dual ternary code is always a polynomial in $\Theta$ and $\Pi^{2}$, where $\Pi = \theta_{0}^{6} - 20 \theta_{0}^{3} \theta_{1}^{3} - 8 \theta_{1}^{6}$.

Now the functions $\theta_{0}$ and $\theta_{1}$ give a mapping of a hyperbolic quotient space for $\Gamma (3)$ to the Riemann sphere. The modular group (quotiented by $\Gamma (3)$) acts on this Riemann sphere by the symmetries of a tetrahedron, where fixed points are the vertices of the tetrahedron.

Note also that the usual generators $S$ and $T$ of the modular group act on polynomials in $\theta_{0}$ and $\theta_{1}$ by simple $2 \times 2$ matrices. In particular, $T$ is given by the phase gate matrix

1 0

0 exp$(\frac{2 \pi i}{3})$

All of this extends to interesting facts about a cube made of two tetrahedrons. Modular forms go from being weight 1 to being weight $\frac{1}{2}$ for the group $\Gamma (4)$. Using $\Gamma (4)$ we obtain the symmetries of a cube from the modular quotient. One can have hours of fun studying Ebeling's course in lattice theory.

Now the functions $\theta_{0}$ and $\theta_{1}$ give a mapping of a hyperbolic quotient space for $\Gamma (3)$ to the Riemann sphere. The modular group (quotiented by $\Gamma (3)$) acts on this Riemann sphere by the symmetries of a tetrahedron, where fixed points are the vertices of the tetrahedron.

Note also that the usual generators $S$ and $T$ of the modular group act on polynomials in $\theta_{0}$ and $\theta_{1}$ by simple $2 \times 2$ matrices. In particular, $T$ is given by the phase gate matrix

1 0

0 exp$(\frac{2 \pi i}{3})$

All of this extends to interesting facts about a cube made of two tetrahedrons. Modular forms go from being weight 1 to being weight $\frac{1}{2}$ for the group $\Gamma (4)$. Using $\Gamma (4)$ we obtain the symmetries of a cube from the modular quotient. One can have hours of fun studying Ebeling's course in lattice theory.

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