from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2289, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([54,18,47]))
chi.galois_orbit()
[g,chi] = znchar(Mod(59,2289))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2289\) | |
| Conductor: | \(2289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(108\) | 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: | $\Q(\zeta_{108})$ |
| Fixed field: | Number field defined by a degree 108 polynomial (not computed) |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2289}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) |
| \(\chi_{2289}(194,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
| \(\chi_{2289}(248,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
| \(\chi_{2289}(290,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) |
| \(\chi_{2289}(341,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
| \(\chi_{2289}(374,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) |
| \(\chi_{2289}(383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) |
| \(\chi_{2289}(446,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
| \(\chi_{2289}(488,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) |
| \(\chi_{2289}(551,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) |
| \(\chi_{2289}(563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) |
| \(\chi_{2289}(584,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) |
| \(\chi_{2289}(614,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) |
| \(\chi_{2289}(752,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) |
| \(\chi_{2289}(803,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) |
| \(\chi_{2289}(866,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
| \(\chi_{2289}(929,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
| \(\chi_{2289}(971,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) |
| \(\chi_{2289}(1025,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) |
| \(\chi_{2289}(1034,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) |
| \(\chi_{2289}(1046,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) |
| \(\chi_{2289}(1076,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) |
| \(\chi_{2289}(1223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{36}\right)\) |
| \(\chi_{2289}(1319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) |
| \(\chi_{2289}(1475,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) |
| \(\chi_{2289}(1487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) |
| \(\chi_{2289}(1508,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) |
| \(\chi_{2289}(1622,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) |
| \(\chi_{2289}(1697,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{36}\right)\) |
| \(\chi_{2289}(1781,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
| \(\chi_{2289}(1823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) |