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Properties

Label 7.256.8.1.1
Class number $1$
Dimension $7$
Determinant $256$
Level $8$

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Lattice Invariants

Dimension:$7$
Determinant:$256$
Level:$8$
Density:$0.107888397114811643230863150904\dots$
Group order:$46080$
Hermite number:$1.35858549639586000739125930982\dots$
Minimal vector length:$3$
Kissing number:$32$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0, 0)$, $(1, 0, 1, -1, 0, 0, 0)$, $(1, 0, 1, -1, 1, -1, 0)$, $(1, 0, 1, 0, 0, -1, 0)$, $(1, 1, 0, -1, 0, 0, 0)$, $(1, 1, 0, 0, -1, 0, 0)$, $(1, 1, 0, 0, 0, -1, 0)$, $(1, 1, 1, -1, 0, -1, 0)$, $(0, 1, -1, 0, -1, 0, 0)$, $(0, 1, 0, 0, 0, 0, 0)$, $(0, 0, 1, -1, 1, 0, 0)$, $(0, 0, 1, 0, 0, 0, 0)$, $(0, 0, 1, 0, 1, -1, 0)$, $(0, 0, 0, 1, 0, 0, 0)$, $(0, 0, 0, 0, 1, 0, 0)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 32 q^{3} \) \(\mathstrut +\mathstrut 62 q^{4} \) \(\mathstrut +\mathstrut 256 q^{7} \) \(\mathstrut +\mathstrut 372 q^{8} \) \(\mathstrut +\mathstrut 864 q^{11} \) \(\mathstrut +\mathstrut 1048 q^{12} \) \(\mathstrut +\mathstrut 1792 q^{15} \) \(\mathstrut +\mathstrut 2110 q^{16} \) \(\mathstrut +\mathstrut 3168 q^{19} \) \(\mathstrut +\mathstrut 3720 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrr} 3 & -1 & -1 & 1 & 1 & 1 & 0 \\ -1 & 3 & 1 & 1 & 1 & 1 & 0 \\ -1 & 1 & 3 & 1 & -1 & 1 & 0 \\ 1 & 1 & 1 & 3 & 1 & 1 & 0 \\ 1 & 1 & -1 & 1 & 3 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrr} 3 & -1 & 1 & 1 & -1 & -1 & 0 \\ -1 & 3 & 1 & 1 & -1 & 1 & 0 \\ 1 & 1 & 3 & 1 & -1 & -1 & 0 \\ 1 & 1 & 1 & 3 & -1 & 1 & 0 \\ -1 & -1 & -1 & -1 & 3 & 1 & 0 \\ -1 & 1 & -1 & 1 & 1 & 3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 4 \end{array}\right)$
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