from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(109, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,109))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(109\) | |
| Conductor: | \(109\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{109}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{83}{108}\right)\) |
| \(\chi_{109}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{23}{108}\right)\) |
| \(\chi_{109}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{85}{108}\right)\) |
| \(\chi_{109}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) |
| \(\chi_{109}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{59}{108}\right)\) |
| \(\chi_{109}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{79}{108}\right)\) |
| \(\chi_{109}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{41}{108}\right)\) |
| \(\chi_{109}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{19}{108}\right)\) |
| \(\chi_{109}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{91}{108}\right)\) |
| \(\chi_{109}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{49}{108}\right)\) |
| \(\chi_{109}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{89}{108}\right)\) |
| \(\chi_{109}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{55}{108}\right)\) |
| \(\chi_{109}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{43}{108}\right)\) |
| \(\chi_{109}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{7}{108}\right)\) |
| \(\chi_{109}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{67}{108}\right)\) |
| \(\chi_{109}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{47}{108}\right)\) |
| \(\chi_{109}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{11}{108}\right)\) |
| \(\chi_{109}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{71}{108}\right)\) |
| \(\chi_{109}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{17}{108}\right)\) |
| \(\chi_{109}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{65}{108}\right)\) |
| \(\chi_{109}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{101}{108}\right)\) |
| \(\chi_{109}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{13}{108}\right)\) |
| \(\chi_{109}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{61}{108}\right)\) |
| \(\chi_{109}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{97}{108}\right)\) |
| \(\chi_{109}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{1}{108}\right)\) |
| \(\chi_{109}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{35}{108}\right)\) |
| \(\chi_{109}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{103}{108}\right)\) |
| \(\chi_{109}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{37}{108}\right)\) |
| \(\chi_{109}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{73}{108}\right)\) |
| \(\chi_{109}(85,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) |
| \(\chi_{109}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{25}{108}\right)\) |