sage: H = DirichletGroup(148)
pari: g = idealstar(,148,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 72 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{36}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{148}(75,\cdot)$, $\chi_{148}(113,\cdot)$ |
First 32 of 72 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{148}(1,\cdot)\) | 148.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{148}(3,\cdot)\) | 148.o | 18 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) |
\(\chi_{148}(5,\cdot)\) | 148.r | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) |
\(\chi_{148}(7,\cdot)\) | 148.p | 18 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) |
\(\chi_{148}(9,\cdot)\) | 148.k | 9 | no | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) |
\(\chi_{148}(11,\cdot)\) | 148.j | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{148}(13,\cdot)\) | 148.r | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{148}(15,\cdot)\) | 148.q | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{148}(17,\cdot)\) | 148.r | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{148}(19,\cdot)\) | 148.q | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{148}(21,\cdot)\) | 148.n | 18 | no | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{148}(23,\cdot)\) | 148.l | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{148}(25,\cdot)\) | 148.n | 18 | no | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{148}(27,\cdot)\) | 148.j | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{148}(29,\cdot)\) | 148.m | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{148}(31,\cdot)\) | 148.g | 4 | yes | \(1\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(1\) | \(1\) | \(-i\) | \(-i\) | \(-i\) | \(i\) | \(-1\) |
\(\chi_{148}(33,\cdot)\) | 148.k | 9 | no | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{148}(35,\cdot)\) | 148.q | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{148}(39,\cdot)\) | 148.q | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) |
\(\chi_{148}(41,\cdot)\) | 148.n | 18 | no | \(1\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) |
\(\chi_{148}(43,\cdot)\) | 148.g | 4 | yes | \(1\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(1\) | \(1\) | \(i\) | \(i\) | \(i\) | \(-i\) | \(-1\) |
\(\chi_{148}(45,\cdot)\) | 148.m | 12 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{148}(47,\cdot)\) | 148.i | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) |
\(\chi_{148}(49,\cdot)\) | 148.k | 9 | no | \(1\) | \(1\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) |
\(\chi_{148}(51,\cdot)\) | 148.l | 12 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{148}(53,\cdot)\) | 148.k | 9 | no | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) |
\(\chi_{148}(55,\cdot)\) | 148.q | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) |
\(\chi_{148}(57,\cdot)\) | 148.r | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) |
\(\chi_{148}(59,\cdot)\) | 148.q | 36 | yes | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) |
\(\chi_{148}(61,\cdot)\) | 148.r | 36 | no | \(-1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) |
\(\chi_{148}(63,\cdot)\) | 148.i | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{148}(65,\cdot)\) | 148.n | 18 | no | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) |