sage: H = DirichletGroup(232)
pari: g = idealstar(,232,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 112 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{2}\times C_{28}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{232}(175,\cdot)$, $\chi_{232}(117,\cdot)$, $\chi_{232}(89,\cdot)$ |
First 32 of 112 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{232}(1,\cdot)\) | 232.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{232}(3,\cdot)\) | 232.v | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(-i\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) |
\(\chi_{232}(5,\cdot)\) | 232.o | 14 | yes | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) |
\(\chi_{232}(7,\cdot)\) | 232.r | 14 | no | \(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) |
\(\chi_{232}(9,\cdot)\) | 232.q | 14 | no | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(-1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) |
\(\chi_{232}(11,\cdot)\) | 232.v | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(-i\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) |
\(\chi_{232}(13,\cdot)\) | 232.o | 14 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
\(\chi_{232}(15,\cdot)\) | 232.x | 28 | no | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(i\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) |
\(\chi_{232}(17,\cdot)\) | 232.j | 4 | no | \(-1\) | \(1\) | \(-i\) | \(-1\) | \(1\) | \(-1\) | \(-i\) | \(-1\) | \(i\) | \(-i\) | \(-i\) | \(-i\) |
\(\chi_{232}(19,\cdot)\) | 232.v | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(-i\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) |
\(\chi_{232}(21,\cdot)\) | 232.u | 28 | yes | \(-1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(-i\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{23}{28}\right)\) |
\(\chi_{232}(23,\cdot)\) | 232.r | 14 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
\(\chi_{232}(25,\cdot)\) | 232.m | 7 | no | \(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) |
\(\chi_{232}(27,\cdot)\) | 232.v | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) |
\(\chi_{232}(31,\cdot)\) | 232.x | 28 | no | \(1\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(-i\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) |
\(\chi_{232}(33,\cdot)\) | 232.q | 14 | no | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(-1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) |
\(\chi_{232}(35,\cdot)\) | 232.t | 14 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
\(\chi_{232}(37,\cdot)\) | 232.u | 28 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(i\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) |
\(\chi_{232}(39,\cdot)\) | 232.x | 28 | no | \(1\) | \(1\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(i\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{27}{28}\right)\) |
\(\chi_{232}(41,\cdot)\) | 232.j | 4 | no | \(-1\) | \(1\) | \(i\) | \(-1\) | \(1\) | \(-1\) | \(i\) | \(-1\) | \(-i\) | \(i\) | \(i\) | \(i\) |
\(\chi_{232}(43,\cdot)\) | 232.v | 28 | yes | \(1\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(-i\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) |
\(\chi_{232}(45,\cdot)\) | 232.s | 14 | yes | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) |
\(\chi_{232}(47,\cdot)\) | 232.x | 28 | no | \(1\) | \(1\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(i\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{19}{28}\right)\) |
\(\chi_{232}(49,\cdot)\) | 232.m | 7 | no | \(1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) |
\(\chi_{232}(51,\cdot)\) | 232.t | 14 | yes | \(-1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) |
\(\chi_{232}(53,\cdot)\) | 232.s | 14 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) |
\(\chi_{232}(55,\cdot)\) | 232.x | 28 | no | \(1\) | \(1\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(i\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) |
\(\chi_{232}(57,\cdot)\) | 232.e | 2 | no | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
\(\chi_{232}(59,\cdot)\) | 232.f | 2 | no | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
\(\chi_{232}(61,\cdot)\) | 232.u | 28 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(-i\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) |
\(\chi_{232}(63,\cdot)\) | 232.n | 14 | no | \(-1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) |
\(\chi_{232}(65,\cdot)\) | 232.m | 7 | no | \(1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) |