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I'm reading now about theta series and its significance in researching self-dual, even lattices. Now, I found the wikipedia article about the Leech lattice, but I'm having trouble understanding where the $\theta$ series have come from.

wikipedia article

As I see it, the $\theta$ series of lattice of order 24, equals to a form in $M_{12} = cE_{12} + \Delta$, so the coefficient doesn't match.

Explanation will be highly appreciated.

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up vote 4 down vote accepted

I'll write down the answer myself, if anyone's interested. The way is to take $\theta_\Lambda = cE_{12} + c'\Delta$. Then by using the fact that $\theta_\Lambda = \sum r_\Lambda (2m)q^m $ we can see, first by checking the zero's coefficients that $c = 1$ and then by taking $m = 1$ that $ 0 = \frac{65520}{691} + c' $ and thus getting what was written in the Wikipedia article.

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