Properties

Label 7.576.18.1.1
Class number $1$
Dimension $7$
Determinant $576$
Level $18$

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Lattice Invariants

Dimension:$7$
Determinant:$576$
Level:$18$
Density:$0.0719255980765410954872421006028\dots$
Group order:$7680$
Hermite number:$1.20997162457437471435521937652\dots$
Minimal vector length:$3$
Kissing number:$20$
Normalized minimal vectors: $(1, -1, 0, -1, 0, 0, 0)$, $(1, 0, 0, 0, 0, 0, 0)$, $(0, 1, 0, 0, 0, 0, 0)$, $(0, 1, 0, 1, 0, 1, 0)$, $(0, 1, 0, 1, 1, 0, 0)$, $(0, 1, 1, 1, 0, 0, 0)$, $(0, 0, 1, 0, 0, 0, 0)$, $(0, 0, 0, 1, 0, 0, 0)$, $(0, 0, 0, 0, 1, 0, 0)$, $(0, 0, 0, 0, 0, 1, 0)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 20 q^{3} \) \(\mathstrut +\mathstrut 42 q^{4} \) \(\mathstrut +\mathstrut 192 q^{7} \) \(\mathstrut +\mathstrut 234 q^{8} \) \(\mathstrut +\mathstrut 468 q^{11} \) \(\mathstrut +\mathstrut 740 q^{12} \) \(\mathstrut +\mathstrut 1440 q^{15} \) \(\mathstrut +\mathstrut 1386 q^{16} \) \(\mathstrut +\mathstrut 1848 q^{19} \) \(\mathstrut +\mathstrut 2340 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrr} 3 & 1 & -1 & 1 & -1 & -1 & 1 \\ 1 & 3 & -1 & -1 & -1 & -1 & -1 \\ -1 & -1 & 3 & -1 & 1 & 1 & 1 \\ 1 & -1 & -1 & 3 & -1 & -1 & 1 \\ -1 & -1 & 1 & -1 & 3 & 1 & -1 \\ -1 & -1 & 1 & -1 & 1 & 3 & 1 \\ 1 & -1 & 1 & 1 & -1 & 1 & 7 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrr} 3 & -1 & 1 & -1 & 1 & -1 & 1 \\ -1 & 3 & -1 & 1 & -1 & -1 & -1 \\ 1 & -1 & 3 & -1 & 1 & -1 & -1 \\ -1 & 1 & -1 & 3 & -1 & 1 & 1 \\ 1 & -1 & 1 & -1 & 3 & -1 & 1 \\ -1 & -1 & -1 & 1 & -1 & 3 & 1 \\ 1 & -1 & -1 & 1 & 1 & 1 & 7 \end{array}\right)$
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