Name: | $C_4$ |
Order: | $4$ |
Abelian: | yes |
Generators: | $\begin{bmatrix}\zeta_8&0&0&0\\0&\zeta_8^7&0&0\\0&0&\zeta_8^7&0\\0&0&0&\zeta_8\end{bmatrix}$ |
$x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
$a_1$ |
$1$ |
$0$ |
$4$ |
$0$ |
$36$ |
$0$ |
$400$ |
$0$ |
$5040$ |
$0$ |
$68544$ |
$0$ |
$975744$ |
$a_2$ |
$1$ |
$2$ |
$8$ |
$32$ |
$150$ |
$732$ |
$3776$ |
$20064$ |
$109318$ |
$605804$ |
$3400848$ |
$19273344$ |
$110017980$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=2\right)\colon$ |
$2$ |
$4$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=4\right)\colon$ |
$8$ |
$16$ |
$36$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=6\right)\colon$ |
$32$ |
$72$ |
$168$ |
$400$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=8\right)\colon$ |
$150$ |
$348$ |
$836$ |
$2040$ |
$5040$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=10\right)\colon$ |
$732$ |
$1768$ |
$4344$ |
$10800$ |
$27104$ |
$68544$ |
$\left(\mathrm{E}\left[a_1^{e_1}a_2^{e_2}\right]:\sum ie_i=12\right)\colon$ |
$3776$ |
$9312$ |
$23272$ |
$58672$ |
$148960$ |
$380352$ |
$975744$ |
$\mathrm{E}\left[\chi_i\chi_j\right] = \begin{bmatrix}1&0&1&2&0&2&0&3&0&2\\0&4&0&0&8&0&8&0&12&0\\1&0&5&6&0&6&0&15&0&12\\2&0&6&12&0&12&0&26&0&18\\0&8&0&0&24&0&24&0&44&0\\2&0&6&12&0&18&0&32&0&24\\0&8&0&0&24&0&28&0&48&0\\3&0&15&26&0&32&0&79&0&62\\0&12&0&0&44&0&48&0&96&0\\2&0&12&18&0&24&0&62&0&54\end{bmatrix}$
$\ \ \ \mathrm{E}\left[\chi_i^2\right] = \begin{bmatrix}1&4&5&12&24&18&28&79&96&54\end{bmatrix}$
| $-$ | $a_2\in\mathbb{Z}$ | $a_2=-2$ | $a_2=-1$ | $a_2=0$ | $a_2=1$ | $a_2=2$ |
---|
$-$ | $1$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|
$a_1=0$ | $1/4$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$ |
---|