from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1009, base_ring=CyclotomicField(252))
M = H._module
chi = DirichletCharacter(H, M([191]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,1009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1009\) | |
| Conductor: | \(1009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(252\) | 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: | $\Q(\zeta_{252})$ |
| Fixed field: | Number field defined by a degree 252 polynomial (not computed) |
First 31 of 72 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1009}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{191}{252}\right)\) |
| \(\chi_{1009}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{247}{252}\right)\) |
| \(\chi_{1009}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{197}{252}\right)\) |
| \(\chi_{1009}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{143}{252}\right)\) |
| \(\chi_{1009}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{199}{252}\right)\) |
| \(\chi_{1009}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{95}{252}\right)\) |
| \(\chi_{1009}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{125}{252}\right)\) |
| \(\chi_{1009}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{109}{252}\right)\) |
| \(\chi_{1009}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{211}{252}\right)\) |
| \(\chi_{1009}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{181}{252}\right)\) |
| \(\chi_{1009}(168,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{131}{252}\right)\) |
| \(\chi_{1009}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{79}{252}\right)\) |
| \(\chi_{1009}(232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{37}{252}\right)\) |
| \(\chi_{1009}(252,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{187}{252}\right)\) |
| \(\chi_{1009}(269,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{179}{252}\right)\) |
| \(\chi_{1009}(274,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{215}{252}\right)\) |
| \(\chi_{1009}(286,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{59}{252}\right)\) |
| \(\chi_{1009}(294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{137}{252}\right)\) |
| \(\chi_{1009}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{101}{252}\right)\) |
| \(\chi_{1009}(324,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{41}{252}\right)\) |
| \(\chi_{1009}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{103}{252}\right)\) |
| \(\chi_{1009}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{13}{252}\right)\) |
| \(\chi_{1009}(402,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{113}{252}\right)\) |
| \(\chi_{1009}(406,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{43}{252}\right)\) |
| \(\chi_{1009}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{155}{252}\right)\) |
| \(\chi_{1009}(411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{19}{252}\right)\) |
| \(\chi_{1009}(420,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{83}{252}\right)\) |
| \(\chi_{1009}(429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{115}{252}\right)\) |
| \(\chi_{1009}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{193}{252}\right)\) |
| \(\chi_{1009}(442,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{173}{252}\right)\) |
| \(\chi_{1009}(444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{157}{252}\right)\) |