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Properties

Label 7.16.8.1.1
Class number $1$
Dimension $7$
Determinant $16$
Level $8$

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Lattice Invariants

Dimension:$7$
Determinant:$16$
Level:$8$
Density:$0.00922805853580351790936795091283\dots$
Group order:$6144$
Hermite number:$0.672950096316178065971416301876\dots$
Minimal vector length:$1$
Kissing number:$4$
Normalized minimal vectors: $(1, 0, 0, 0, 0, 0, 0)$, $(0, 1, 0, 0, 0, 0, 0)$
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Theta Series

\(1 \) \(\mathstrut +\mathstrut 4 q \) \(\mathstrut +\mathstrut 12 q^{2} \) \(\mathstrut +\mathstrut 64 q^{3} \) \(\mathstrut +\mathstrut 190 q^{4} \) \(\mathstrut +\mathstrut 240 q^{5} \) \(\mathstrut +\mathstrut 184 q^{6} \) \(\mathstrut +\mathstrut 512 q^{7} \) \(\mathstrut +\mathstrut 1140 q^{8} \) \(\mathstrut +\mathstrut 1012 q^{9} \) \(\mathstrut +\mathstrut 632 q^{10} \) \(\mathstrut +\mathstrut 1728 q^{11} \) \(\mathstrut +\mathstrut 3096 q^{12} \) \(\mathstrut +\mathstrut 2416 q^{13} \) \(\mathstrut +\mathstrut 1584 q^{14} \) \(\mathstrut +\mathstrut 3584 q^{15} \) \(\mathstrut +\mathstrut 6206 q^{16} \) \(\mathstrut +\mathstrut 5088 q^{17} \) \(\mathstrut +\mathstrut 2820 q^{18} \) \(\mathstrut +\mathstrut 6336 q^{19} \) \(\mathstrut +\mathstrut 11400 q^{20} \) \(\mathstrut +\mathstrut O(q^{21}) \)

Gram Matrix

$\left(\begin{array}{rrrrrrr} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 3 & 1 & -1 & -1 \\ 0 & 0 & 0 & 1 & 2 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 2 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 & 2 \end{array}\right)$

Genus Structure

Genus representatives:
Class number:$1$
 
$\left(\begin{array}{rrrrrrr} 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 2 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 2 & -1 & -1 \\ 0 & 0 & 1 & 1 & -1 & 3 & 2 \\ 0 & 0 & 1 & 1 & -1 & 2 & 3 \end{array}\right)$
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