sage: H = DirichletGroup(47)
pari: g = idealstar(,47,2)
Character group
sage: G.order()
pari: g.no
| ||
Order | = | 46 |
sage: H.invariants()
pari: g.cyc
| ||
Structure | = | \(C_{46}\) |
sage: H.gens()
pari: g.gen
| ||
Generators | = | $\chi_{47}(5,\cdot)$ |
First 32 of 46 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{47}(1,\cdot)\) | 47.a | 1 | no | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) | \(1\) |
\(\chi_{47}(2,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) |
\(\chi_{47}(3,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) |
\(\chi_{47}(4,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) |
\(\chi_{47}(5,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{7}{46}\right)\) |
\(\chi_{47}(6,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) |
\(\chi_{47}(7,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) |
\(\chi_{47}(8,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) |
\(\chi_{47}(9,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) |
\(\chi_{47}(10,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{41}{46}\right)\) |
\(\chi_{47}(11,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{3}{46}\right)\) |
\(\chi_{47}(12,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) |
\(\chi_{47}(13,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{31}{46}\right)\) |
\(\chi_{47}(14,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) |
\(\chi_{47}(15,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{9}{46}\right)\) |
\(\chi_{47}(16,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) |
\(\chi_{47}(17,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) |
\(\chi_{47}(18,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
\(\chi_{47}(19,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{39}{46}\right)\) |
\(\chi_{47}(20,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{29}{46}\right)\) |
\(\chi_{47}(21,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) |
\(\chi_{47}(22,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{37}{46}\right)\) |
\(\chi_{47}(23,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{35}{46}\right)\) |
\(\chi_{47}(24,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) |
\(\chi_{47}(25,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) |
\(\chi_{47}(26,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{19}{46}\right)\) |
\(\chi_{47}(27,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) |
\(\chi_{47}(28,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) |
\(\chi_{47}(29,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{15}{46}\right)\) |
\(\chi_{47}(30,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{43}{46}\right)\) |
\(\chi_{47}(31,\cdot)\) | 47.d | 46 | yes | \(-1\) | \(1\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{21}{46}\right)\) |
\(\chi_{47}(32,\cdot)\) | 47.c | 23 | yes | \(1\) | \(1\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) |