from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1009, base_ring=CyclotomicField(504))
M = H._module
chi = DirichletCharacter(H, M([443]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1009))
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: | \(504\) | 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: | $\Q(\zeta_{504})$ |
| Fixed field: | Number field defined by a degree 504 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1009}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{252}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{1}{252}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{443}{504}\right)\) |
| \(\chi_{1009}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{252}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{205}{252}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{347}{504}\right)\) |
| \(\chi_{1009}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{252}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{59}{252}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{433}{504}\right)\) |
| \(\chi_{1009}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{252}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{115}{252}\right)\) | \(e\left(\frac{47}{126}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{41}{504}\right)\) |
| \(\chi_{1009}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{252}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{191}{252}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{445}{504}\right)\) |
| \(\chi_{1009}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{252}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{223}{252}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{257}{504}\right)\) |
| \(\chi_{1009}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{252}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{11}{252}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{337}{504}\right)\) |
| \(\chi_{1009}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{252}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{5}{252}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{199}{504}\right)\) |
| \(\chi_{1009}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{252}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{67}{252}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{449}{504}\right)\) |
| \(\chi_{1009}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{181}{252}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{61}{252}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{311}{504}\right)\) |
| \(\chi_{1009}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{252}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{137}{252}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{211}{504}\right)\) |
| \(\chi_{1009}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{252}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{67}{126}\right)\) | \(e\left(\frac{127}{252}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{317}{504}\right)\) |
| \(\chi_{1009}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{252}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{215}{252}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{241}{504}\right)\) |
| \(\chi_{1009}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{252}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{209}{252}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{103}{504}\right)\) |
| \(\chi_{1009}(84,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{252}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{193}{252}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{323}{504}\right)\) |
| \(\chi_{1009}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{252}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{17}{252}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{223}{504}\right)\) |
| \(\chi_{1009}(108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{252}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{173}{252}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{31}{504}\right)\) |
| \(\chi_{1009}(120,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{241}{252}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{13}{252}\right)\) | \(e\left(\frac{59}{126}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{126}\right)\) | \(e\left(\frac{215}{504}\right)\) |
| \(\chi_{1009}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{252}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{95}{252}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{1}{504}\right)\) |
| \(\chi_{1009}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{252}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{155}{252}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{121}{504}\right)\) |
| \(\chi_{1009}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{252}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{83}{126}\right)\) | \(e\left(\frac{131}{252}\right)\) | \(e\left(\frac{13}{126}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{73}{504}\right)\) |
| \(\chi_{1009}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{252}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{25}{252}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{491}{504}\right)\) |
| \(\chi_{1009}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{252}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{89}{252}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{115}{504}\right)\) |
| \(\chi_{1009}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{169}{252}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{43}{126}\right)\) | \(e\left(\frac{121}{252}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{179}{504}\right)\) |
| \(\chi_{1009}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{252}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{73}{252}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{335}{504}\right)\) |
| \(\chi_{1009}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{252}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{37}{252}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{263}{504}\right)\) |
| \(\chi_{1009}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{252}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{179}{252}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{421}{504}\right)\) |
| \(\chi_{1009}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{252}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{229}{252}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{73}{126}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{143}{504}\right)\) |
| \(\chi_{1009}(174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{252}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{53}{126}\right)\) | \(e\left(\frac{29}{252}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{247}{504}\right)\) |
| \(\chi_{1009}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{252}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{17}{126}\right)\) | \(e\left(\frac{83}{252}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{5}{126}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{113}{126}\right)\) | \(e\left(\frac{229}{504}\right)\) |
| \(\chi_{1009}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{252}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{53}{252}\right)\) | \(e\left(\frac{37}{126}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{101}{126}\right)\) | \(e\left(\frac{43}{504}\right)\) |