| Name: | $\mathrm{U}(1)$ |
| Index: | $2$ |
| $\mathbb{R}$-dimension: | $1$ |
| Description: | $\left\{\begin{bmatrix}\alpha&0\\0&\bar\alpha\end{bmatrix}:\alpha\bar\alpha = 1,\ \alpha\in\mathbb{C}\right\}$ |
| $x$ |
$\mathrm{E}[x^{0}]$ |
$\mathrm{E}[x^{1}]$ |
$\mathrm{E}[x^{2}]$ |
$\mathrm{E}[x^{3}]$ |
$\mathrm{E}[x^{4}]$ |
$\mathrm{E}[x^{5}]$ |
$\mathrm{E}[x^{6}]$ |
$\mathrm{E}[x^{7}]$ |
$\mathrm{E}[x^{8}]$ |
$\mathrm{E}[x^{9}]$ |
$\mathrm{E}[x^{10}]$ |
$\mathrm{E}[x^{11}]$ |
$\mathrm{E}[x^{12}]$ |
| $a_1$ |
$1$ |
$0$ |
$1$ |
$0$ |
$3$ |
$0$ |
$10$ |
$0$ |
$35$ |
$0$ |
$126$ |
$0$ |
$462$ |
| $\mathrm{P}[a_1=0]=\frac{1}{2}$ |
Additional information
$N(\mathrm{U}(1)$ is the Sato-Tate group of any elliptic curve with complex multiplication defined over a number field that does not contain the CM field; the Sato-Tate conjecture in this case follows from a classical results of Deuring; see Lemma 2.15 of [arXiv:1604.01256].
