sage: H = DirichletGroup(116)
pari: g = idealstar(,116,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 56 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{28}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{116}(59,\cdot)$, $\chi_{116}(89,\cdot)$ |
First 32 of 56 characters
Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.
Character | Orbit | Order | Primitive | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{116}(1,\cdot)\) | 116.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{116}(3,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) |
\(\chi_{116}(5,\cdot)\) | 116.i | 14 | no | \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) |
\(\chi_{116}(7,\cdot)\) | 116.j | 14 | yes | \(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) |
\(\chi_{116}(9,\cdot)\) | 116.i | 14 | no | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(-1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) |
\(\chi_{116}(11,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{28}\right)\) |
\(\chi_{116}(13,\cdot)\) | 116.i | 14 | no | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) |
\(\chi_{116}(15,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(i\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) |
\(\chi_{116}(17,\cdot)\) | 116.f | 4 | no | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(1\) | \(-1\) | \(-i\) | \(-1\) | \(i\) | \(-i\) | \(-i\) | \(-i\) |
\(\chi_{116}(19,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) |
\(\chi_{116}(21,\cdot)\) | 116.k | 28 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-i\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) |
\(\chi_{116}(23,\cdot)\) | 116.j | 14 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
\(\chi_{116}(25,\cdot)\) | 116.g | 7 | no | \(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) |
\(\chi_{116}(27,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(i\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{28}\right)\) |
\(\chi_{116}(31,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) |
\(\chi_{116}(33,\cdot)\) | 116.i | 14 | no | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(-1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) |
\(\chi_{116}(35,\cdot)\) | 116.h | 14 | yes | \(-1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) |
\(\chi_{116}(37,\cdot)\) | 116.k | 28 | no | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(i\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) |
\(\chi_{116}(39,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(i\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) |
\(\chi_{116}(41,\cdot)\) | 116.f | 4 | no | \(-1\) | \(1\) | \(i\) | \(-1\) | \(1\) | \(-1\) | \(i\) | \(-1\) | \(-i\) | \(i\) | \(i\) | \(i\) |
\(\chi_{116}(43,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-i\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) |
\(\chi_{116}(45,\cdot)\) | 116.g | 7 | no | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
\(\chi_{116}(47,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) |
\(\chi_{116}(49,\cdot)\) | 116.g | 7 | no | \(1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) |
\(\chi_{116}(51,\cdot)\) | 116.h | 14 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) |
\(\chi_{116}(53,\cdot)\) | 116.g | 7 | no | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) |
\(\chi_{116}(55,\cdot)\) | 116.l | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(i\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) |
\(\chi_{116}(57,\cdot)\) | 116.c | 2 | no | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
\(\chi_{116}(59,\cdot)\) | 116.b | 2 | no | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) |
\(\chi_{116}(61,\cdot)\) | 116.k | 28 | no | \(-1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(-i\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) |
\(\chi_{116}(63,\cdot)\) | 116.h | 14 | yes | \(-1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) |
\(\chi_{116}(65,\cdot)\) | 116.g | 7 | no | \(1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) |