from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5265, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([70,81,36]))
chi.galois_orbit()
[g,chi] = znchar(Mod(68,5265))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5265\) | |
| Conductor: | \(5265\) | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(17\) | \(19\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5265}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{53}{108}\right)\) |
| \(\chi_{5265}(302,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{59}{108}\right)\) |
| \(\chi_{5265}(308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{13}{108}\right)\) |
| \(\chi_{5265}(542,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{55}{108}\right)\) |
| \(\chi_{5265}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{41}{108}\right)\) |
| \(\chi_{5265}(887,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{47}{108}\right)\) |
| \(\chi_{5265}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{37}{108}\right)\) |
| \(\chi_{5265}(1127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{79}{108}\right)\) |
| \(\chi_{5265}(1238,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{29}{108}\right)\) |
| \(\chi_{5265}(1472,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{35}{108}\right)\) |
| \(\chi_{5265}(1478,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{61}{108}\right)\) |
| \(\chi_{5265}(1712,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{103}{108}\right)\) |
| \(\chi_{5265}(1823,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{108}\right)\) |
| \(\chi_{5265}(2057,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{108}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{23}{108}\right)\) |
| \(\chi_{5265}(2063,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{85}{108}\right)\) |
| \(\chi_{5265}(2297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{19}{108}\right)\) |
| \(\chi_{5265}(2408,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{108}\right)\) |
| \(\chi_{5265}(2642,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{108}\right)\) |
| \(\chi_{5265}(2648,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{108}\right)\) |
| \(\chi_{5265}(2882,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{43}{108}\right)\) |
| \(\chi_{5265}(2993,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{101}{108}\right)\) |
| \(\chi_{5265}(3227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{107}{108}\right)\) |
| \(\chi_{5265}(3233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{25}{108}\right)\) |
| \(\chi_{5265}(3467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{67}{108}\right)\) |
| \(\chi_{5265}(3578,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{89}{108}\right)\) |
| \(\chi_{5265}(3812,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{95}{108}\right)\) |
| \(\chi_{5265}(3818,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{49}{108}\right)\) |
| \(\chi_{5265}(4052,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{91}{108}\right)\) |
| \(\chi_{5265}(4163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{77}{108}\right)\) |
| \(\chi_{5265}(4397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{83}{108}\right)\) |
| \(\chi_{5265}(4403,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{73}{108}\right)\) |