from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([4,63]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,2009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2009\) | |
| Conductor: | \(2009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(168\) | 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: | $\Q(\zeta_{168})$ |
| Fixed field: | Number field defined by a degree 168 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2009}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{65}{168}\right)\) |
| \(\chi_{2009}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{97}{168}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{101}{168}\right)\) |
| \(\chi_{2009}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{107}{168}\right)\) | \(e\left(\frac{31}{168}\right)\) |
| \(\chi_{2009}(150,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{107}{168}\right)\) |
| \(\chi_{2009}(208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{101}{168}\right)\) | \(e\left(\frac{1}{168}\right)\) |
| \(\chi_{2009}(243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{157}{168}\right)\) |
| \(\chi_{2009}(290,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{41}{168}\right)\) |
| \(\chi_{2009}(355,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{43}{168}\right)\) |
| \(\chi_{2009}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{151}{168}\right)\) |
| \(\chi_{2009}(437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{151}{168}\right)\) | \(e\left(\frac{83}{168}\right)\) |
| \(\chi_{2009}(465,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{95}{168}\right)\) |
| \(\chi_{2009}(495,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{121}{168}\right)\) |
| \(\chi_{2009}(530,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{109}{168}\right)\) |
| \(\chi_{2009}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{17}{168}\right)\) |
| \(\chi_{2009}(612,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{53}{168}\right)\) |
| \(\chi_{2009}(642,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{163}{168}\right)\) |
| \(\chi_{2009}(670,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{103}{168}\right)\) |
| \(\chi_{2009}(724,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{79}{168}\right)\) | \(e\left(\frac{59}{168}\right)\) |
| \(\chi_{2009}(752,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{71}{168}\right)\) |
| \(\chi_{2009}(782,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{73}{168}\right)\) |
| \(\chi_{2009}(817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{61}{168}\right)\) |
| \(\chi_{2009}(899,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{29}{168}\right)\) |
| \(\chi_{2009}(929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{115}{168}\right)\) |
| \(\chi_{2009}(957,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{55}{168}\right)\) |
| \(\chi_{2009}(1039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{47}{168}\right)\) |
| \(\chi_{2009}(1069,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{5}{168}\right)\) | \(e\left(\frac{25}{168}\right)\) |
| \(\chi_{2009}(1104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{13}{168}\right)\) |
| \(\chi_{2009}(1151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{137}{168}\right)\) |
| \(\chi_{2009}(1186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{5}{168}\right)\) |
| \(\chi_{2009}(1216,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{67}{168}\right)\) |
| \(\chi_{2009}(1298,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{11}{168}\right)\) |