sage: H = DirichletGroup(74)
pari: g = idealstar(,74,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 36 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{36}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{74}(39,\cdot)$ |
First 32 of 36 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{74}(1,\cdot)\) | 74.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{74}(3,\cdot)\) | 74.h | 18 | no | \(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{74}(5,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) |
\(\chi_{74}(7,\cdot)\) | 74.f | 9 | no | \(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{74}(9,\cdot)\) | 74.f | 9 | no | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{74}(11,\cdot)\) | 74.e | 6 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{74}(13,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{74}(15,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{74}(17,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{74}(19,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{74}(21,\cdot)\) | 74.h | 18 | no | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{74}(23,\cdot)\) | 74.g | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{74}(25,\cdot)\) | 74.h | 18 | no | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{74}(27,\cdot)\) | 74.e | 6 | no | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{74}(29,\cdot)\) | 74.g | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{74}(31,\cdot)\) | 74.d | 4 | no | \(-1\) | \(1\) | \(-1\) | \(-i\) | \(1\) | \(1\) | \(-1\) | \(-i\) | \(i\) | \(-i\) | \(-i\) | \(-1\) |
\(\chi_{74}(33,\cdot)\) | 74.f | 9 | no | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{74}(35,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{74}(39,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{74}(41,\cdot)\) | 74.h | 18 | no | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{74}(43,\cdot)\) | 74.d | 4 | no | \(-1\) | \(1\) | \(-1\) | \(i\) | \(1\) | \(1\) | \(-1\) | \(i\) | \(-i\) | \(i\) | \(i\) | \(-1\) |
\(\chi_{74}(45,\cdot)\) | 74.g | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{74}(47,\cdot)\) | 74.c | 3 | no | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{74}(49,\cdot)\) | 74.f | 9 | no | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{74}(51,\cdot)\) | 74.g | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{74}(53,\cdot)\) | 74.f | 9 | no | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{74}(55,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{74}(57,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{74}(59,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{74}(61,\cdot)\) | 74.i | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{74}(63,\cdot)\) | 74.c | 3 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{74}(65,\cdot)\) | 74.h | 18 | no | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) |