sage: H = DirichletGroup(77)
pari: g = idealstar(,77,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 60 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{2}\times C_{30}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{77}(45,\cdot)$, $\chi_{77}(57,\cdot)$ |
First 32 of 60 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{77}(1,\cdot)\) | 77.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{77}(2,\cdot)\) | 77.o | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{77}(3,\cdot)\) | 77.p | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{10}\right)\) |
\(\chi_{77}(4,\cdot)\) | 77.m | 15 | yes | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) |
\(\chi_{77}(5,\cdot)\) | 77.p | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{77}(6,\cdot)\) | 77.l | 10 | yes | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{2}{5}\right)\) |
\(\chi_{77}(8,\cdot)\) | 77.k | 10 | no | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) |
\(\chi_{77}(9,\cdot)\) | 77.m | 15 | yes | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{5}\right)\) |
\(\chi_{77}(10,\cdot)\) | 77.i | 6 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) |
\(\chi_{77}(12,\cdot)\) | 77.g | 6 | no | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) |
\(\chi_{77}(13,\cdot)\) | 77.l | 10 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{3}{5}\right)\) |
\(\chi_{77}(15,\cdot)\) | 77.f | 5 | no | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) |
\(\chi_{77}(16,\cdot)\) | 77.m | 15 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{5}\right)\) |
\(\chi_{77}(17,\cdot)\) | 77.n | 30 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) |
\(\chi_{77}(18,\cdot)\) | 77.o | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{10}\right)\) |
\(\chi_{77}(19,\cdot)\) | 77.n | 30 | yes | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) |
\(\chi_{77}(20,\cdot)\) | 77.j | 10 | yes | \(-1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{77}(23,\cdot)\) | 77.e | 3 | no | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) |
\(\chi_{77}(24,\cdot)\) | 77.n | 30 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{5}\right)\) |
\(\chi_{77}(25,\cdot)\) | 77.m | 15 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{5}\right)\) |
\(\chi_{77}(26,\cdot)\) | 77.p | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{10}\right)\) |
\(\chi_{77}(27,\cdot)\) | 77.j | 10 | yes | \(-1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{77}(29,\cdot)\) | 77.k | 10 | no | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) |
\(\chi_{77}(30,\cdot)\) | 77.o | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{3}{10}\right)\) |
\(\chi_{77}(31,\cdot)\) | 77.p | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{77}(32,\cdot)\) | 77.h | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) |
\(\chi_{77}(34,\cdot)\) | 77.d | 2 | no | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
\(\chi_{77}(36,\cdot)\) | 77.f | 5 | no | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) |
\(\chi_{77}(37,\cdot)\) | 77.m | 15 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{5}\right)\) |
\(\chi_{77}(38,\cdot)\) | 77.p | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{77}(39,\cdot)\) | 77.o | 30 | yes | \(-1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{77}(40,\cdot)\) | 77.n | 30 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{5}\right)\) |