from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([33,67]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,4033))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4033\) | |
| Conductor: | \(4033\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4033}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(1\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{71}{108}\right)\) |
| \(\chi_{4033}(204,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(1\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{23}{108}\right)\) |
| \(\chi_{4033}(257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(1\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{31}{108}\right)\) |
| \(\chi_{4033}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{108}\right)\) | \(e\left(\frac{35}{108}\right)\) |
| \(\chi_{4033}(392,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{79}{108}\right)\) |
| \(\chi_{4033}(426,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{67}{108}\right)\) | \(e\left(\frac{95}{108}\right)\) |
| \(\chi_{4033}(505,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{108}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{1}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) |
| \(\chi_{4033}(575,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{1}{108}\right)\) |
| \(\chi_{4033}(930,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(1\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{11}{108}\right)\) |
| \(\chi_{4033}(1053,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{19}{108}\right)\) |
| \(\chi_{4033}(1142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{29}{108}\right)\) |
| \(\chi_{4033}(1241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(1\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{17}{108}\right)\) | \(e\left(\frac{37}{108}\right)\) |
| \(\chi_{4033}(1297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(1\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{41}{108}\right)\) | \(e\left(\frac{13}{108}\right)\) |
| \(\chi_{4033}(1367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{108}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(1\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{103}{108}\right)\) |
| \(\chi_{4033}(1411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{43}{108}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{47}{108}\right)\) |
| \(\chi_{4033}(1720,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{85}{108}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{5}{108}\right)\) |
| \(\chi_{4033}(1791,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{29}{108}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{97}{108}\right)\) |
| \(\chi_{4033}(1835,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(1\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{47}{108}\right)\) | \(e\left(\frac{7}{108}\right)\) |
| \(\chi_{4033}(2198,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{65}{108}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(1\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{101}{108}\right)\) | \(e\left(\frac{61}{108}\right)\) |
| \(\chi_{4033}(2242,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{83}{108}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{108}\right)\) | \(e\left(\frac{43}{108}\right)\) |
| \(\chi_{4033}(2313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(1\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{103}{108}\right)\) | \(e\left(\frac{59}{108}\right)\) |
| \(\chi_{4033}(2622,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{61}{108}\right)\) | \(e\left(\frac{101}{108}\right)\) |
| \(\chi_{4033}(2666,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{77}{108}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(1\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{5}{108}\right)\) | \(e\left(\frac{49}{108}\right)\) |
| \(\chi_{4033}(2736,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{59}{108}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(1\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{95}{108}\right)\) | \(e\left(\frac{67}{108}\right)\) |
| \(\chi_{4033}(2792,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{35}{108}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(1\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{91}{108}\right)\) |
| \(\chi_{4033}(2891,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{108}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(1\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{79}{108}\right)\) | \(e\left(\frac{83}{108}\right)\) |
| \(\chi_{4033}(2980,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(1\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{89}{108}\right)\) | \(e\left(\frac{73}{108}\right)\) |
| \(\chi_{4033}(3103,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{25}{108}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(1\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{97}{108}\right)\) | \(e\left(\frac{65}{108}\right)\) |
| \(\chi_{4033}(3458,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{71}{108}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{107}{108}\right)\) | \(e\left(\frac{55}{108}\right)\) |
| \(\chi_{4033}(3528,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{91}{108}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(1\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{55}{108}\right)\) | \(e\left(\frac{107}{108}\right)\) |
| \(\chi_{4033}(3607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{49}{108}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(1\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{108}\right)\) | \(e\left(\frac{41}{108}\right)\) |