from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3971, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([0,47]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,3971))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3971\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(114\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 361.j | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3971}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{5}{38}\right)\) |
| \(\chi_{3971}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{13}{38}\right)\) |
| \(\chi_{3971}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{31}{38}\right)\) |
| \(\chi_{3971}(331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{33}{38}\right)\) |
| \(\chi_{3971}(540,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{15}{38}\right)\) |
| \(\chi_{3971}(639,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{7}{38}\right)\) |
| \(\chi_{3971}(749,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{35}{38}\right)\) |
| \(\chi_{3971}(848,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{33}{38}\right)\) |
| \(\chi_{3971}(958,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{17}{38}\right)\) |
| \(\chi_{3971}(1057,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{21}{38}\right)\) |
| \(\chi_{3971}(1167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{37}{38}\right)\) |
| \(\chi_{3971}(1266,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{9}{38}\right)\) |
| \(\chi_{3971}(1475,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{35}{38}\right)\) |
| \(\chi_{3971}(1585,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{1}{38}\right)\) |
| \(\chi_{3971}(1684,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{23}{38}\right)\) |
| \(\chi_{3971}(1794,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{21}{38}\right)\) |
| \(\chi_{3971}(1893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{11}{38}\right)\) |
| \(\chi_{3971}(2003,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{3}{38}\right)\) |
| \(\chi_{3971}(2102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{37}{38}\right)\) |
| \(\chi_{3971}(2212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{23}{38}\right)\) |
| \(\chi_{3971}(2311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) |
| \(\chi_{3971}(2421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{5}{38}\right)\) |
| \(\chi_{3971}(2520,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{13}{38}\right)\) |
| \(\chi_{3971}(2630,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{25}{38}\right)\) |
| \(\chi_{3971}(2729,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{1}{38}\right)\) |
| \(\chi_{3971}(2839,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{7}{38}\right)\) |
| \(\chi_{3971}(2938,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{27}{38}\right)\) |
| \(\chi_{3971}(3048,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{27}{38}\right)\) |
| \(\chi_{3971}(3147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{15}{38}\right)\) |
| \(\chi_{3971}(3257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{9}{38}\right)\) |
| \(\chi_{3971}(3356,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{3}{38}\right)\) |