Name: | $C_4$ |
Order: | $4$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}0&0&0&\zeta_8\\0&0&-\zeta_8^7&0\\0&-\zeta_8^7&0&0\\\zeta_8&0&0&0\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$2$ |
$0$ |
$24$ |
$0$ |
$320$ |
$0$ |
$4480$ |
$0$ |
$64512$ |
$0$ |
$946176$ |
$a_2$ |
$1$ |
$1$ |
$5$ |
$22$ |
$115$ |
$606$ |
$3314$ |
$18348$ |
$102883$ |
$581494$ |
$3308470$ |
$18920628$ |
$108665902$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$1$ |
$2$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$5$ |
$10$ |
$24$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$22$ |
$52$ |
$128$ |
$320$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$115$ |
$278$ |
$696$ |
$1760$ |
$4480$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$606$ |
$1516$ |
$3840$ |
$9792$ |
$25088$ |
$64512$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$3314$ |
$8388$ |
$21424$ |
$54976$ |
$141568$ |
$365568$ |
$946176$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&0&1&0&2&0&3&0&2\\0&2&0&0&6&0&6&0&12&0\\0&0&4&4&0&5&0&13&0&12\\1&0&4&9&0&11&0&24&0&18\\0&6&0&0&20&0&22&0&42&0\\2&0&5&11&0&16&0&30&0&23\\0&6&0&0&22&0&26&0&48&0\\3&0&13&24&0&30&0&77&0&61\\0&12&0&0&42&0&48&0&92&0\\2&0&12&18&0&23&0&61&0&52\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&2&4&9&20&16&26&77&92&52\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
---|
$-$ | $1$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
---|
$a_1=0$ | $3/4$ | $1/2$ | $0$ | $0$ | $1/2$ | $0$ | $0$ |
---|