Name: | $C_4$ |
Order: | $4$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8^7\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$12$ |
$0$ |
$100$ |
$0$ |
$1008$ |
$0$ |
$11424$ |
$0$ |
$139392$ |
$a_2$ |
$1$ |
$1$ |
$4$ |
$11$ |
$46$ |
$182$ |
$824$ |
$3817$ |
$18582$ |
$92678$ |
$473368$ |
$2458326$ |
$12947532$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$4$ |
$6$ |
$12$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$11$ |
$22$ |
$46$ |
$100$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$46$ |
$90$ |
$196$ |
$440$ |
$1008$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$182$ |
$396$ |
$892$ |
$2056$ |
$4816$ |
$11424$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$824$ |
$1836$ |
$4248$ |
$10000$ |
$23840$ |
$57408$ |
$139392$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&1&0&0&0&0\\0&2&0&0&2&0&2&0&2&0\\0&0&3&1&0&0&0&4&0&5\\1&0&1&4&0&3&0&4&0&1\\0&2&0&0&6&0&4&0&8&0\\1&0&0&3&0&8&0&3&0&0\\0&2&0&0&4&0&6&0&6&0\\0&0&4&4&0&3&0&15&0&12\\0&2&0&0&8&0&6&0&16&0\\0&0&5&1&0&0&0&12&0&16\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&3&4&6&8&6&15&16&16\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|