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Properties

Label 7.16.32.1.1
Class number $1$
Dimension $7$
Determinant $16$
Level $32$

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Lattice Invariants

Dimension:$7$
Determinant:$16$
Level:$32$
Density:$0.104403564285649124355987973892\dots$
Group order:$10080$
Hermite number:$1.34590019263235613194283260375\dots$
Minimal vector length:$2$
Kissing number:$42$
Normalized minimal vectors: $(1, -1, 0, 0, 0, 0, 0)$, $(1, 0, -1, 0, 0, 0, 0)$, $(1, 0, 0, 0, -1, 0, 0)$, $(1, 0, 0, 0, 0, -1, 0)$, $(1, 0, 0, 0, 0, 0, 0)$, $(1, 0, 0, 1, 0, 0, 0)$, $(0, 1, -1, 0, 0, 0, 0)$, $(0, 1, 0, 0, -1, 0, 0)$, $(0, 1, 0, 0, 0, -1, 0)$, $(0, 1, 0, 0, 0, 0, 0)$, $(0, 1, 0, 1, 0, 0, 0)$, $(0, 0, 1, 0, -1, 0, 0)$, $(0, 0, 1, 0, 0, -1, 0)$, $(0, 0, 1, 0, 0, 0, 0)$, $(0, 0, 1, 1, 0, 0, 0)$ ...
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 42 q^{2} \) \(\mathstrut +\mathstrut 280 q^{4} \) \(\mathstrut +\mathstrut 644 q^{6} \) \(\mathstrut +\mathstrut 1764 q^{8} \) \(\mathstrut +\mathstrut 2212 q^{10} \) \(\mathstrut +\mathstrut 4480 q^{12} \) \(\mathstrut +\mathstrut 5544 q^{14} \) \(\mathstrut +\mathstrut 9534 q^{16} \) \(\mathstrut +\mathstrut 9870 q^{18} \) \(\mathstrut +\mathstrut 16800 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrr} 2 & 1 & 1 & -1 & 1 & 1 & 1 \\ 1 & 2 & 1 & -1 & 1 & 1 & 0 \\ 1 & 1 & 2 & -1 & 1 & 1 & 1 \\ -1 & -1 & -1 & 2 & -1 & -1 & 0 \\ 1 & 1 & 1 & -1 & 2 & 1 & 1 \\ 1 & 1 & 1 & -1 & 1 & 2 & 0 \\ 1 & 0 & 1 & 0 & 1 & 0 & 4 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrr} 2 & -1 & 1 & 1 & 1 & 1 & -1 \\ -1 & 2 & -1 & -1 & -1 & -1 & 1 \\ 1 & -1 & 2 & 1 & 1 & 1 & 0 \\ 1 & -1 & 1 & 2 & 1 & 1 & -1 \\ 1 & -1 & 1 & 1 & 2 & 1 & 0 \\ 1 & -1 & 1 & 1 & 1 & 2 & -1 \\ -1 & 1 & 0 & -1 & 0 & -1 & 4 \end{array}\right)$
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