from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([57,38,89]))
chi.galois_orbit()
[g,chi] = znchar(Mod(107,7581))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7581\) | |
| Conductor: | \(7581\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(114\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{57})$ |
| Fixed field: | Number field defined by a degree 114 polynomial (not computed) |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7581}(107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{7}{38}\right)\) |
| \(\chi_{7581}(179,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{35}{38}\right)\) |
| \(\chi_{7581}(506,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{33}{38}\right)\) |
| \(\chi_{7581}(578,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{17}{38}\right)\) |
| \(\chi_{7581}(905,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{21}{38}\right)\) |
| \(\chi_{7581}(977,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{37}{38}\right)\) |
| \(\chi_{7581}(1304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{9}{38}\right)\) |
| \(\chi_{7581}(1703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{35}{38}\right)\) |
| \(\chi_{7581}(1775,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{1}{38}\right)\) |
| \(\chi_{7581}(2102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{23}{38}\right)\) |
| \(\chi_{7581}(2174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{21}{38}\right)\) |
| \(\chi_{7581}(2501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{11}{38}\right)\) |
| \(\chi_{7581}(2573,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{3}{38}\right)\) |
| \(\chi_{7581}(2900,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{37}{38}\right)\) |
| \(\chi_{7581}(2972,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{23}{38}\right)\) |
| \(\chi_{7581}(3299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) |
| \(\chi_{7581}(3371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{5}{38}\right)\) |
| \(\chi_{7581}(3698,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{13}{38}\right)\) |
| \(\chi_{7581}(3770,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) |
| \(\chi_{7581}(4097,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{1}{38}\right)\) |
| \(\chi_{7581}(4169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{7}{38}\right)\) |
| \(\chi_{7581}(4496,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{27}{38}\right)\) |
| \(\chi_{7581}(4568,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{27}{38}\right)\) |
| \(\chi_{7581}(4895,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{15}{38}\right)\) |
| \(\chi_{7581}(4967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{9}{38}\right)\) |
| \(\chi_{7581}(5294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{3}{38}\right)\) |
| \(\chi_{7581}(5366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{29}{38}\right)\) |
| \(\chi_{7581}(5693,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{29}{38}\right)\) |
| \(\chi_{7581}(5765,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{11}{38}\right)\) |
| \(\chi_{7581}(6092,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{17}{38}\right)\) |
| \(\chi_{7581}(6164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{31}{38}\right)\) |