sage: H = DirichletGroup(67)
pari: g = idealstar(,67,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 66 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{66}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{67}(2,\cdot)$ |
First 32 of 66 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{67}(1,\cdot)\) | 67.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{67}(2,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) |
\(\chi_{67}(3,\cdot)\) | 67.f | 22 | yes | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) |
\(\chi_{67}(4,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) |
\(\chi_{67}(5,\cdot)\) | 67.f | 22 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) |
\(\chi_{67}(6,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) |
\(\chi_{67}(7,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) |
\(\chi_{67}(8,\cdot)\) | 67.f | 22 | yes | \(-1\) | \(1\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) |
\(\chi_{67}(9,\cdot)\) | 67.e | 11 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) |
\(\chi_{67}(10,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) |
\(\chi_{67}(11,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) |
\(\chi_{67}(12,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{43}{66}\right)\) |
\(\chi_{67}(13,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) |
\(\chi_{67}(14,\cdot)\) | 67.e | 11 | yes | \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
\(\chi_{67}(15,\cdot)\) | 67.e | 11 | yes | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) |
\(\chi_{67}(16,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) |
\(\chi_{67}(17,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) |
\(\chi_{67}(18,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{41}{66}\right)\) |
\(\chi_{67}(19,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) |
\(\chi_{67}(20,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) |
\(\chi_{67}(21,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) |
\(\chi_{67}(22,\cdot)\) | 67.e | 11 | yes | \(1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) |
\(\chi_{67}(23,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) |
\(\chi_{67}(24,\cdot)\) | 67.e | 11 | yes | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
\(\chi_{67}(25,\cdot)\) | 67.e | 11 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) |
\(\chi_{67}(26,\cdot)\) | 67.g | 33 | yes | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) |
\(\chi_{67}(27,\cdot)\) | 67.f | 22 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) |
\(\chi_{67}(28,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) |
\(\chi_{67}(29,\cdot)\) | 67.c | 3 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
\(\chi_{67}(30,\cdot)\) | 67.d | 6 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |
\(\chi_{67}(31,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{1}{66}\right)\) |
\(\chi_{67}(32,\cdot)\) | 67.h | 66 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) |