from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(840))
M = H._module
chi = DirichletCharacter(H, M([800,63]))
chi.galois_orbit()
[g,chi] = znchar(Mod(11,2009))
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: | \(840\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{840})$ |
| Fixed field: | Number field defined by a degree 840 polynomial (not computed) |
First 31 of 192 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2009}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{420}\right)\) | \(e\left(\frac{13}{168}\right)\) | \(e\left(\frac{89}{210}\right)\) | \(e\left(\frac{113}{420}\right)\) | \(e\left(\frac{221}{280}\right)\) | \(e\left(\frac{19}{140}\right)\) | \(e\left(\frac{13}{84}\right)\) | \(e\left(\frac{103}{105}\right)\) | \(e\left(\frac{269}{840}\right)\) | \(e\left(\frac{421}{840}\right)\) |
| \(\chi_{2009}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{420}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{101}{210}\right)\) | \(e\left(\frac{317}{420}\right)\) | \(e\left(\frac{29}{280}\right)\) | \(e\left(\frac{31}{140}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{52}{105}\right)\) | \(e\left(\frac{461}{840}\right)\) | \(e\left(\frac{709}{840}\right)\) |
| \(\chi_{2009}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{289}{420}\right)\) | \(e\left(\frac{71}{168}\right)\) | \(e\left(\frac{79}{210}\right)\) | \(e\left(\frac{223}{420}\right)\) | \(e\left(\frac{31}{280}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{23}{105}\right)\) | \(e\left(\frac{319}{840}\right)\) | \(e\left(\frac{671}{840}\right)\) |
| \(\chi_{2009}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{420}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{47}{210}\right)\) | \(e\left(\frac{239}{420}\right)\) | \(e\left(\frac{263}{280}\right)\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{19}{105}\right)\) | \(e\left(\frac{647}{840}\right)\) | \(e\left(\frac{463}{840}\right)\) |
| \(\chi_{2009}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{420}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{139}{210}\right)\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{51}{280}\right)\) | \(e\left(\frac{69}{140}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{83}{105}\right)\) | \(e\left(\frac{19}{840}\right)\) | \(e\left(\frac{11}{840}\right)\) |
| \(\chi_{2009}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{420}\right)\) | \(e\left(\frac{31}{168}\right)\) | \(e\left(\frac{83}{210}\right)\) | \(e\left(\frac{11}{420}\right)\) | \(e\left(\frac{247}{280}\right)\) | \(e\left(\frac{13}{140}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{76}{105}\right)\) | \(e\left(\frac{383}{840}\right)\) | \(e\left(\frac{487}{840}\right)\) |
| \(\chi_{2009}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{420}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{169}{210}\right)\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{61}{280}\right)\) | \(e\left(\frac{99}{140}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{8}{105}\right)\) | \(e\left(\frac{709}{840}\right)\) | \(e\left(\frac{101}{840}\right)\) |
| \(\chi_{2009}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{420}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{113}{210}\right)\) | \(e\left(\frac{101}{420}\right)\) | \(e\left(\frac{257}{280}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{233}{840}\right)\) | \(e\left(\frac{577}{840}\right)\) |
| \(\chi_{2009}(130,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{347}{420}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{89}{420}\right)\) | \(e\left(\frac{153}{280}\right)\) | \(e\left(\frac{67}{140}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{4}{105}\right)\) | \(e\left(\frac{617}{840}\right)\) | \(e\left(\frac{313}{840}\right)\) |
| \(\chi_{2009}(135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{420}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{151}{210}\right)\) | \(e\left(\frac{397}{420}\right)\) | \(e\left(\frac{69}{280}\right)\) | \(e\left(\frac{11}{140}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{32}{105}\right)\) | \(e\left(\frac{421}{840}\right)\) | \(e\left(\frac{509}{840}\right)\) |
| \(\chi_{2009}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{420}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{127}{210}\right)\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{103}{280}\right)\) | \(e\left(\frac{57}{140}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{29}{105}\right)\) | \(e\left(\frac{247}{840}\right)\) | \(e\left(\frac{143}{840}\right)\) |
| \(\chi_{2009}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{61}{210}\right)\) | \(e\left(\frac{127}{420}\right)\) | \(e\left(\frac{39}{280}\right)\) | \(e\left(\frac{61}{140}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{47}{105}\right)\) | \(e\left(\frac{31}{840}\right)\) | \(e\left(\frac{239}{840}\right)\) |
| \(\chi_{2009}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{420}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{143}{210}\right)\) | \(e\left(\frac{401}{420}\right)\) | \(e\left(\frac{197}{280}\right)\) | \(e\left(\frac{3}{140}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{31}{105}\right)\) | \(e\left(\frac{293}{840}\right)\) | \(e\left(\frac{37}{840}\right)\) |
| \(\chi_{2009}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{420}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{173}{210}\right)\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{67}{280}\right)\) | \(e\left(\frac{33}{140}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{61}{105}\right)\) | \(e\left(\frac{563}{840}\right)\) | \(e\left(\frac{547}{840}\right)\) |
| \(\chi_{2009}(170,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{420}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{331}{420}\right)\) | \(e\left(\frac{127}{280}\right)\) | \(e\left(\frac{73}{140}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{101}{105}\right)\) | \(e\left(\frac{223}{840}\right)\) | \(e\left(\frac{527}{840}\right)\) |
| \(\chi_{2009}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{420}\right)\) | \(e\left(\frac{163}{168}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{47}{420}\right)\) | \(e\left(\frac{139}{280}\right)\) | \(e\left(\frac{81}{140}\right)\) | \(e\left(\frac{79}{84}\right)\) | \(e\left(\frac{67}{105}\right)\) | \(e\left(\frac{491}{840}\right)\) | \(e\left(\frac{19}{840}\right)\) |
| \(\chi_{2009}(186,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{377}{420}\right)\) | \(e\left(\frac{115}{168}\right)\) | \(e\left(\frac{167}{210}\right)\) | \(e\left(\frac{179}{420}\right)\) | \(e\left(\frac{163}{280}\right)\) | \(e\left(\frac{97}{140}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{467}{840}\right)\) | \(e\left(\frac{403}{840}\right)\) |
| \(\chi_{2009}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{89}{280}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{401}{840}\right)\) | \(e\left(\frac{409}{840}\right)\) |
| \(\chi_{2009}(198,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{420}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{169}{420}\right)\) | \(e\left(\frac{53}{280}\right)\) | \(e\left(\frac{47}{140}\right)\) | \(e\left(\frac{41}{84}\right)\) | \(e\left(\frac{89}{105}\right)\) | \(e\left(\frac{157}{840}\right)\) | \(e\left(\frac{533}{840}\right)\) |
| \(\chi_{2009}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{420}\right)\) | \(e\left(\frac{17}{168}\right)\) | \(e\left(\frac{97}{210}\right)\) | \(e\left(\frac{109}{420}\right)\) | \(e\left(\frac{233}{280}\right)\) | \(e\left(\frac{27}{140}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{104}{105}\right)\) | \(e\left(\frac{817}{840}\right)\) | \(e\left(\frac{473}{840}\right)\) |
| \(\chi_{2009}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{403}{420}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{193}{210}\right)\) | \(e\left(\frac{61}{420}\right)\) | \(e\left(\frac{237}{280}\right)\) | \(e\left(\frac{123}{140}\right)\) | \(e\left(\frac{65}{84}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{253}{840}\right)\) | \(e\left(\frac{677}{840}\right)\) |
| \(\chi_{2009}(235,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{419}{420}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{209}{210}\right)\) | \(e\left(\frac{53}{420}\right)\) | \(e\left(\frac{121}{280}\right)\) | \(e\left(\frac{139}{140}\right)\) | \(e\left(\frac{73}{84}\right)\) | \(e\left(\frac{13}{105}\right)\) | \(e\left(\frac{89}{840}\right)\) | \(e\left(\frac{361}{840}\right)\) |
| \(\chi_{2009}(240,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{420}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{31}{420}\right)\) | \(e\left(\frac{187}{280}\right)\) | \(e\left(\frac{113}{140}\right)\) | \(e\left(\frac{11}{84}\right)\) | \(e\left(\frac{71}{105}\right)\) | \(e\left(\frac{163}{840}\right)\) | \(e\left(\frac{227}{840}\right)\) |
| \(\chi_{2009}(261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{420}\right)\) | \(e\left(\frac{59}{168}\right)\) | \(e\left(\frac{181}{210}\right)\) | \(e\left(\frac{67}{420}\right)\) | \(e\left(\frac{219}{280}\right)\) | \(e\left(\frac{41}{140}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{62}{105}\right)\) | \(e\left(\frac{691}{840}\right)\) | \(e\left(\frac{179}{840}\right)\) |
| \(\chi_{2009}(268,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{420}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{79}{420}\right)\) | \(e\left(\frac{43}{280}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{59}{105}\right)\) | \(e\left(\frac{307}{840}\right)\) | \(e\left(\frac{443}{840}\right)\) |
| \(\chi_{2009}(270,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{420}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{149}{210}\right)\) | \(e\left(\frac{83}{420}\right)\) | \(e\left(\frac{171}{280}\right)\) | \(e\left(\frac{9}{140}\right)\) | \(e\left(\frac{43}{84}\right)\) | \(e\left(\frac{58}{105}\right)\) | \(e\left(\frac{179}{840}\right)\) | \(e\left(\frac{811}{840}\right)\) |
| \(\chi_{2009}(298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{420}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{59}{210}\right)\) | \(e\left(\frac{233}{420}\right)\) | \(e\left(\frac{141}{280}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{629}{840}\right)\) | \(e\left(\frac{541}{840}\right)\) |
| \(\chi_{2009}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{420}\right)\) | \(e\left(\frac{89}{168}\right)\) | \(e\left(\frac{199}{210}\right)\) | \(e\left(\frac{373}{420}\right)\) | \(e\left(\frac{1}{280}\right)\) | \(e\left(\frac{59}{140}\right)\) | \(e\left(\frac{5}{84}\right)\) | \(e\left(\frac{38}{105}\right)\) | \(e\left(\frac{769}{840}\right)\) | \(e\left(\frac{401}{840}\right)\) |
| \(\chi_{2009}(340,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{420}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{229}{280}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{22}{105}\right)\) | \(e\left(\frac{821}{840}\right)\) | \(e\left(\frac{829}{840}\right)\) |
| \(\chi_{2009}(345,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{420}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{19}{210}\right)\) | \(e\left(\frac{43}{420}\right)\) | \(e\left(\frac{151}{280}\right)\) | \(e\left(\frac{89}{140}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{199}{840}\right)\) | \(e\left(\frac{71}{840}\right)\) |
| \(\chi_{2009}(347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{420}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{17}{210}\right)\) | \(e\left(\frac{359}{420}\right)\) | \(e\left(\frac{183}{280}\right)\) | \(e\left(\frac{17}{140}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{94}{105}\right)\) | \(e\left(\frac{167}{840}\right)\) | \(e\left(\frac{583}{840}\right)\) |