from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13132, base_ring=CyclotomicField(462))
M = H._module
chi = DirichletCharacter(H, M([0,319,105]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,13132))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(13132\) | |
| Conductor: | \(3283\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(462\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 3283.cw | 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: | $\Q(\zeta_{231})$ |
| Fixed field: | Number field defined by a degree 462 polynomial (not computed) |
First 31 of 120 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{13132}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{128}{231}\right)\) | \(e\left(\frac{100}{231}\right)\) | \(e\left(\frac{25}{231}\right)\) | \(e\left(\frac{13}{462}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{373}{462}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{139}{231}\right)\) | \(e\left(\frac{200}{231}\right)\) |
| \(\chi_{13132}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{160}{231}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{89}{231}\right)\) | \(e\left(\frac{305}{462}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{293}{462}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{19}{231}\right)\) |
| \(\chi_{13132}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{231}\right)\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{218}{231}\right)\) | \(e\left(\frac{215}{462}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{305}{462}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{131}{231}\right)\) | \(e\left(\frac{127}{231}\right)\) |
| \(\chi_{13132}(745,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{231}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{38}{231}\right)\) | \(e\left(\frac{29}{462}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{299}{462}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{8}{231}\right)\) | \(e\left(\frac{73}{231}\right)\) |
| \(\chi_{13132}(789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{231}\right)\) | \(e\left(\frac{184}{231}\right)\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{181}{462}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) |
| \(\chi_{13132}(857,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{130}{231}\right)\) | \(e\left(\frac{116}{231}\right)\) | \(e\left(\frac{29}{231}\right)\) | \(e\left(\frac{89}{462}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{137}{462}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{152}{231}\right)\) | \(e\left(\frac{1}{231}\right)\) |
| \(\chi_{13132}(929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{164}{231}\right)\) | \(e\left(\frac{157}{231}\right)\) | \(e\left(\frac{97}{231}\right)\) | \(e\left(\frac{457}{462}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{283}{462}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{83}{231}\right)\) |
| \(\chi_{13132}(941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{231}\right)\) | \(e\left(\frac{194}{231}\right)\) | \(e\left(\frac{164}{231}\right)\) | \(e\left(\frac{113}{462}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{257}{462}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{157}{231}\right)\) |
| \(\chi_{13132}(1013,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{337}{462}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{145}{462}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{85}{231}\right)\) | \(e\left(\frac{227}{231}\right)\) |
| \(\chi_{13132}(1125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{231}\right)\) | \(e\left(\frac{94}{231}\right)\) | \(e\left(\frac{139}{231}\right)\) | \(e\left(\frac{331}{462}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{115}{462}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{188}{231}\right)\) |
| \(\chi_{13132}(1181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{212}{231}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{193}{231}\right)\) | \(e\left(\frac{433}{462}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{163}{462}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{158}{231}\right)\) |
| \(\chi_{13132}(1209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{40}{231}\right)\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{421}{462}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{103}{462}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{80}{231}\right)\) |
| \(\chi_{13132}(1249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{231}\right)\) | \(e\left(\frac{137}{231}\right)\) | \(e\left(\frac{92}{231}\right)\) | \(e\left(\frac{131}{462}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{347}{462}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{43}{231}\right)\) |
| \(\chi_{13132}(1517,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{231}\right)\) | \(e\left(\frac{115}{231}\right)\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{373}{462}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{325}{462}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{79}{231}\right)\) | \(e\left(\frac{230}{231}\right)\) |
| \(\chi_{13132}(1613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{231}\right)\) | \(e\left(\frac{188}{231}\right)\) | \(e\left(\frac{47}{231}\right)\) | \(e\left(\frac{431}{462}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{461}{462}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{145}{231}\right)\) |
| \(\chi_{13132}(1769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{231}\right)\) | \(e\left(\frac{142}{231}\right)\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{97}{462}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{331}{462}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{53}{231}\right)\) |
| \(\chi_{13132}(1921,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{231}\right)\) | \(e\left(\frac{26}{231}\right)\) | \(e\left(\frac{122}{231}\right)\) | \(e\left(\frac{239}{462}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{425}{462}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{50}{231}\right)\) | \(e\left(\frac{52}{231}\right)\) |
| \(\chi_{13132}(2189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{231}\right)\) | \(e\left(\frac{103}{231}\right)\) | \(e\left(\frac{199}{231}\right)\) | \(e\left(\frac{85}{462}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{271}{462}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{206}{231}\right)\) |
| \(\chi_{13132}(2397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{231}\right)\) | \(e\left(\frac{41}{231}\right)\) | \(e\left(\frac{68}{231}\right)\) | \(e\left(\frac{137}{462}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{377}{462}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{221}{231}\right)\) | \(e\left(\frac{82}{231}\right)\) |
| \(\chi_{13132}(2537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{80}{231}\right)\) | \(e\left(\frac{20}{231}\right)\) | \(e\left(\frac{149}{462}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{437}{462}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{65}{231}\right)\) | \(e\left(\frac{160}{231}\right)\) |
| \(\chi_{13132}(2621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{231}\right)\) | \(e\left(\frac{53}{231}\right)\) | \(e\left(\frac{71}{231}\right)\) | \(e\left(\frac{425}{462}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{431}{462}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{173}{231}\right)\) | \(e\left(\frac{106}{231}\right)\) |
| \(\chi_{13132}(2733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{17}{231}\right)\) | \(e\left(\frac{62}{231}\right)\) | \(e\left(\frac{23}{462}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{269}{462}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{86}{231}\right)\) | \(e\left(\frac{34}{231}\right)\) |
| \(\chi_{13132}(2789,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{231}\right)\) | \(e\left(\frac{167}{231}\right)\) | \(e\left(\frac{215}{231}\right)\) | \(e\left(\frac{389}{462}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{251}{462}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{179}{231}\right)\) | \(e\left(\frac{103}{231}\right)\) |
| \(\chi_{13132}(2805,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{190}{231}\right)\) | \(e\left(\frac{163}{231}\right)\) | \(e\left(\frac{325}{462}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{85}{462}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{10}{231}\right)\) | \(e\left(\frac{149}{231}\right)\) |
| \(\chi_{13132}(2817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{214}{231}\right)\) | \(e\left(\frac{95}{231}\right)\) | \(e\left(\frac{197}{231}\right)\) | \(e\left(\frac{47}{462}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{389}{462}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{5}{231}\right)\) | \(e\left(\frac{190}{231}\right)\) |
| \(\chi_{13132}(2889,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{206}{231}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{181}{231}\right)\) | \(e\left(\frac{205}{462}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{409}{462}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{184}{231}\right)\) | \(e\left(\frac{62}{231}\right)\) |
| \(\chi_{13132}(3001,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{218}{231}\right)\) | \(e\left(\frac{127}{231}\right)\) | \(e\left(\frac{205}{231}\right)\) | \(e\left(\frac{199}{462}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{379}{462}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{31}{231}\right)\) | \(e\left(\frac{23}{231}\right)\) |
| \(\chi_{13132}(3085,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{231}\right)\) | \(e\left(\frac{73}{231}\right)\) | \(e\left(\frac{76}{231}\right)\) | \(e\left(\frac{289}{462}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{367}{462}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{16}{231}\right)\) | \(e\left(\frac{146}{231}\right)\) |
| \(\chi_{13132}(3125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{38}{231}\right)\) | \(e\left(\frac{125}{231}\right)\) | \(e\left(\frac{65}{462}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{17}{462}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{2}{231}\right)\) | \(e\left(\frac{76}{231}\right)\) |
| \(\chi_{13132}(3377,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{202}{231}\right)\) | \(e\left(\frac{230}{231}\right)\) | \(e\left(\frac{173}{231}\right)\) | \(e\left(\frac{53}{462}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{419}{462}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{158}{231}\right)\) | \(e\left(\frac{229}{231}\right)\) |
| \(\chi_{13132}(3393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{134}{231}\right)\) | \(e\left(\frac{148}{231}\right)\) | \(e\left(\frac{37}{231}\right)\) | \(e\left(\frac{241}{462}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{127}{462}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{178}{231}\right)\) | \(e\left(\frac{65}{231}\right)\) |