from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6480, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([0,27,88,54]))
chi.galois_orbit()
[g,chi] = znchar(Mod(229,6480))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6480\) | |
| Conductor: | \(6480\) | 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: | even | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6480}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{37}{54}\right)\) |
| \(\chi_{6480}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{35}{54}\right)\) |
| \(\chi_{6480}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{13}{54}\right)\) |
| \(\chi_{6480}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{29}{54}\right)\) |
| \(\chi_{6480}(949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{43}{54}\right)\) |
| \(\chi_{6480}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{23}{54}\right)\) |
| \(\chi_{6480}(1309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{19}{54}\right)\) |
| \(\chi_{6480}(1429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{17}{54}\right)\) |
| \(\chi_{6480}(1669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{49}{54}\right)\) |
| \(\chi_{6480}(1789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{54}\right)\) |
| \(\chi_{6480}(2029,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{25}{54}\right)\) |
| \(\chi_{6480}(2149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{54}\right)\) |
| \(\chi_{6480}(2389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{54}\right)\) |
| \(\chi_{6480}(2509,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{53}{54}\right)\) |
| \(\chi_{6480}(2749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{31}{54}\right)\) |
| \(\chi_{6480}(2869,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{47}{54}\right)\) |
| \(\chi_{6480}(3109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{54}\right)\) |
| \(\chi_{6480}(3229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{41}{54}\right)\) |
| \(\chi_{6480}(3469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{37}{54}\right)\) |
| \(\chi_{6480}(3589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{35}{54}\right)\) |
| \(\chi_{6480}(3829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{13}{54}\right)\) |
| \(\chi_{6480}(3949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{29}{54}\right)\) |
| \(\chi_{6480}(4189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{43}{54}\right)\) |
| \(\chi_{6480}(4309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{54}\right)\) |
| \(\chi_{6480}(4549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{19}{54}\right)\) |
| \(\chi_{6480}(4669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{54}\right)\) |
| \(\chi_{6480}(4909,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{49}{54}\right)\) |
| \(\chi_{6480}(5029,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{11}{54}\right)\) |
| \(\chi_{6480}(5269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{25}{54}\right)\) |
| \(\chi_{6480}(5389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{5}{54}\right)\) |
| \(\chi_{6480}(5629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{54}\right)\) |