sage: H = DirichletGroup(155)
pari: g = idealstar(,155,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 120 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{60}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{155}(32,\cdot)$, $\chi_{155}(96,\cdot)$ |
First 32 of 120 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{155}(1,\cdot)\) | 155.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{155}(2,\cdot)\) | 155.s | 20 | yes | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) |
\(\chi_{155}(3,\cdot)\) | 155.x | 60 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) |
\(\chi_{155}(4,\cdot)\) | 155.n | 10 | yes | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{155}(6,\cdot)\) | 155.k | 6 | no | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{155}(7,\cdot)\) | 155.w | 60 | yes | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) |
\(\chi_{155}(8,\cdot)\) | 155.s | 20 | yes | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) |
\(\chi_{155}(9,\cdot)\) | 155.u | 30 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) |
\(\chi_{155}(11,\cdot)\) | 155.t | 30 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) |
\(\chi_{155}(12,\cdot)\) | 155.x | 60 | yes | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) |
\(\chi_{155}(13,\cdot)\) | 155.x | 60 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) |
\(\chi_{155}(14,\cdot)\) | 155.u | 30 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) |
\(\chi_{155}(16,\cdot)\) | 155.h | 5 | no | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) |
\(\chi_{155}(17,\cdot)\) | 155.x | 60 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) |
\(\chi_{155}(18,\cdot)\) | 155.w | 60 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) |
\(\chi_{155}(19,\cdot)\) | 155.u | 30 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) |
\(\chi_{155}(21,\cdot)\) | 155.t | 30 | no | \(-1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) |
\(\chi_{155}(22,\cdot)\) | 155.x | 60 | yes | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) |
\(\chi_{155}(23,\cdot)\) | 155.r | 20 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) |
\(\chi_{155}(24,\cdot)\) | 155.v | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) |
\(\chi_{155}(26,\cdot)\) | 155.k | 6 | no | \(-1\) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) |
\(\chi_{155}(27,\cdot)\) | 155.r | 20 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) |
\(\chi_{155}(28,\cdot)\) | 155.w | 60 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) |
\(\chi_{155}(29,\cdot)\) | 155.m | 10 | yes | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) |
\(\chi_{155}(32,\cdot)\) | 155.g | 4 | no | \(-1\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(-i\) |
\(\chi_{155}(33,\cdot)\) | 155.s | 20 | yes | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) |
\(\chi_{155}(34,\cdot)\) | 155.v | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) |
\(\chi_{155}(36,\cdot)\) | 155.e | 3 | no | \(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{155}(37,\cdot)\) | 155.p | 12 | yes | \(1\) | \(1\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-i\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) |
\(\chi_{155}(38,\cdot)\) | 155.w | 60 | yes | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) |
\(\chi_{155}(39,\cdot)\) | 155.n | 10 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{155}(41,\cdot)\) | 155.q | 15 | no | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) |