from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(114))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(8,361))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(361\) | |
| Conductor: | \(361\) | 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: | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{361}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) |
| \(\chi_{361}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) |
| \(\chi_{361}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) |
| \(\chi_{361}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{7}{19}\right)\) |
| \(\chi_{361}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) |
| \(\chi_{361}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{113}{114}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) |
| \(\chi_{361}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) |
| \(\chi_{361}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) |
| \(\chi_{361}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{6}{19}\right)\) |
| \(\chi_{361}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{35}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) |
| \(\chi_{361}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) |
| \(\chi_{361}(122,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{91}{114}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{14}{19}\right)\) |
| \(\chi_{361}(126,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{31}{114}\right)\) | \(e\left(\frac{18}{19}\right)\) |
| \(\chi_{361}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{7}{114}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) |
| \(\chi_{361}(145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{5}{19}\right)\) |
| \(\chi_{361}(160,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{13}{19}\right)\) |
| \(\chi_{361}(164,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{11}{19}\right)\) |
| \(\chi_{361}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) |
| \(\chi_{361}(183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{17}{19}\right)\) |
| \(\chi_{361}(198,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{12}{19}\right)\) |
| \(\chi_{361}(202,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{114}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{4}{19}\right)\) |
| \(\chi_{361}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{37}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{2}{19}\right)\) |
| \(\chi_{361}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) |
| \(\chi_{361}(236,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{11}{19}\right)\) |
| \(\chi_{361}(240,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{16}{19}\right)\) |
| \(\chi_{361}(255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{1}{19}\right)\) |
| \(\chi_{361}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{3}{19}\right)\) |
| \(\chi_{361}(274,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{103}{114}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{89}{114}\right)\) | \(e\left(\frac{10}{19}\right)\) |
| \(\chi_{361}(278,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{114}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) |
| \(\chi_{361}(297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{67}{114}\right)\) | \(e\left(\frac{15}{19}\right)\) |
| \(\chi_{361}(312,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{114}\right)\) | \(e\left(\frac{49}{114}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{101}{114}\right)\) | \(e\left(\frac{9}{19}\right)\) |