from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5269, base_ring=CyclotomicField(478))
M = H._module
chi = DirichletCharacter(H, M([0,121]))
chi.galois_orbit()
[g,chi] = znchar(Mod(34,5269))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5269\) | |
| Conductor: | \(479\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(478\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 479.d | 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: | $\Q(\zeta_{239})$ |
| Fixed field: | Number field defined by a degree 478 polynomial (not computed) |
First 31 of 238 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5269}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{152}{239}\right)\) | \(e\left(\frac{215}{239}\right)\) | \(e\left(\frac{65}{239}\right)\) | \(e\left(\frac{220}{239}\right)\) | \(e\left(\frac{128}{239}\right)\) | \(e\left(\frac{32}{239}\right)\) | \(e\left(\frac{217}{239}\right)\) | \(e\left(\frac{191}{239}\right)\) | \(e\left(\frac{133}{239}\right)\) | \(e\left(\frac{41}{239}\right)\) |
| \(\chi_{5269}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{156}{239}\right)\) | \(e\left(\frac{76}{239}\right)\) | \(e\left(\frac{73}{239}\right)\) | \(e\left(\frac{100}{239}\right)\) | \(e\left(\frac{232}{239}\right)\) | \(e\left(\frac{58}{239}\right)\) | \(e\left(\frac{229}{239}\right)\) | \(e\left(\frac{152}{239}\right)\) | \(e\left(\frac{17}{239}\right)\) | \(e\left(\frac{149}{239}\right)\) |
| \(\chi_{5269}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{239}\right)\) | \(e\left(\frac{201}{239}\right)\) | \(e\left(\frac{83}{239}\right)\) | \(e\left(\frac{189}{239}\right)\) | \(e\left(\frac{123}{239}\right)\) | \(e\left(\frac{210}{239}\right)\) | \(e\left(\frac{5}{239}\right)\) | \(e\left(\frac{163}{239}\right)\) | \(e\left(\frac{111}{239}\right)\) | \(e\left(\frac{45}{239}\right)\) |
| \(\chi_{5269}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{239}\right)\) | \(e\left(\frac{233}{239}\right)\) | \(e\left(\frac{76}{239}\right)\) | \(e\left(\frac{55}{239}\right)\) | \(e\left(\frac{32}{239}\right)\) | \(e\left(\frac{8}{239}\right)\) | \(e\left(\frac{114}{239}\right)\) | \(e\left(\frac{227}{239}\right)\) | \(e\left(\frac{93}{239}\right)\) | \(e\left(\frac{70}{239}\right)\) |
| \(\chi_{5269}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{76}{239}\right)\) | \(e\left(\frac{227}{239}\right)\) | \(e\left(\frac{152}{239}\right)\) | \(e\left(\frac{110}{239}\right)\) | \(e\left(\frac{64}{239}\right)\) | \(e\left(\frac{16}{239}\right)\) | \(e\left(\frac{228}{239}\right)\) | \(e\left(\frac{215}{239}\right)\) | \(e\left(\frac{186}{239}\right)\) | \(e\left(\frac{140}{239}\right)\) |
| \(\chi_{5269}(155,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{87}{239}\right)\) | \(e\left(\frac{24}{239}\right)\) | \(e\left(\frac{174}{239}\right)\) | \(e\left(\frac{19}{239}\right)\) | \(e\left(\frac{111}{239}\right)\) | \(e\left(\frac{207}{239}\right)\) | \(e\left(\frac{22}{239}\right)\) | \(e\left(\frac{48}{239}\right)\) | \(e\left(\frac{106}{239}\right)\) | \(e\left(\frac{198}{239}\right)\) |
| \(\chi_{5269}(166,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{239}\right)\) | \(e\left(\frac{79}{239}\right)\) | \(e\left(\frac{35}{239}\right)\) | \(e\left(\frac{192}{239}\right)\) | \(e\left(\frac{216}{239}\right)\) | \(e\left(\frac{54}{239}\right)\) | \(e\left(\frac{172}{239}\right)\) | \(e\left(\frac{158}{239}\right)\) | \(e\left(\frac{90}{239}\right)\) | \(e\left(\frac{114}{239}\right)\) |
| \(\chi_{5269}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{70}{239}\right)\) | \(e\left(\frac{77}{239}\right)\) | \(e\left(\frac{140}{239}\right)\) | \(e\left(\frac{51}{239}\right)\) | \(e\left(\frac{147}{239}\right)\) | \(e\left(\frac{216}{239}\right)\) | \(e\left(\frac{210}{239}\right)\) | \(e\left(\frac{154}{239}\right)\) | \(e\left(\frac{121}{239}\right)\) | \(e\left(\frac{217}{239}\right)\) |
| \(\chi_{5269}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{48}{239}\right)\) | \(e\left(\frac{5}{239}\right)\) | \(e\left(\frac{96}{239}\right)\) | \(e\left(\frac{233}{239}\right)\) | \(e\left(\frac{53}{239}\right)\) | \(e\left(\frac{73}{239}\right)\) | \(e\left(\frac{144}{239}\right)\) | \(e\left(\frac{10}{239}\right)\) | \(e\left(\frac{42}{239}\right)\) | \(e\left(\frac{101}{239}\right)\) |
| \(\chi_{5269}(199,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{218}{239}\right)\) | \(e\left(\frac{192}{239}\right)\) | \(e\left(\frac{197}{239}\right)\) | \(e\left(\frac{152}{239}\right)\) | \(e\left(\frac{171}{239}\right)\) | \(e\left(\frac{222}{239}\right)\) | \(e\left(\frac{176}{239}\right)\) | \(e\left(\frac{145}{239}\right)\) | \(e\left(\frac{131}{239}\right)\) | \(e\left(\frac{150}{239}\right)\) |
| \(\chi_{5269}(232,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{239}\right)\) | \(e\left(\frac{97}{239}\right)\) | \(e\left(\frac{46}{239}\right)\) | \(e\left(\frac{27}{239}\right)\) | \(e\left(\frac{120}{239}\right)\) | \(e\left(\frac{30}{239}\right)\) | \(e\left(\frac{69}{239}\right)\) | \(e\left(\frac{194}{239}\right)\) | \(e\left(\frac{50}{239}\right)\) | \(e\left(\frac{143}{239}\right)\) |
| \(\chi_{5269}(254,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{118}{239}\right)\) | \(e\left(\frac{82}{239}\right)\) | \(e\left(\frac{236}{239}\right)\) | \(e\left(\frac{45}{239}\right)\) | \(e\left(\frac{200}{239}\right)\) | \(e\left(\frac{50}{239}\right)\) | \(e\left(\frac{115}{239}\right)\) | \(e\left(\frac{164}{239}\right)\) | \(e\left(\frac{163}{239}\right)\) | \(e\left(\frac{79}{239}\right)\) |
| \(\chi_{5269}(265,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{239}\right)\) | \(e\left(\frac{122}{239}\right)\) | \(e\left(\frac{48}{239}\right)\) | \(e\left(\frac{236}{239}\right)\) | \(e\left(\frac{146}{239}\right)\) | \(e\left(\frac{156}{239}\right)\) | \(e\left(\frac{72}{239}\right)\) | \(e\left(\frac{5}{239}\right)\) | \(e\left(\frac{21}{239}\right)\) | \(e\left(\frac{170}{239}\right)\) |
| \(\chi_{5269}(287,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{160}{239}\right)\) | \(e\left(\frac{176}{239}\right)\) | \(e\left(\frac{81}{239}\right)\) | \(e\left(\frac{219}{239}\right)\) | \(e\left(\frac{97}{239}\right)\) | \(e\left(\frac{84}{239}\right)\) | \(e\left(\frac{2}{239}\right)\) | \(e\left(\frac{113}{239}\right)\) | \(e\left(\frac{140}{239}\right)\) | \(e\left(\frac{18}{239}\right)\) |
| \(\chi_{5269}(298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{239}\right)\) | \(e\left(\frac{108}{239}\right)\) | \(e\left(\frac{66}{239}\right)\) | \(e\left(\frac{205}{239}\right)\) | \(e\left(\frac{141}{239}\right)\) | \(e\left(\frac{95}{239}\right)\) | \(e\left(\frac{99}{239}\right)\) | \(e\left(\frac{216}{239}\right)\) | \(e\left(\frac{238}{239}\right)\) | \(e\left(\frac{174}{239}\right)\) |
| \(\chi_{5269}(342,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{239}\right)\) | \(e\left(\frac{125}{239}\right)\) | \(e\left(\frac{10}{239}\right)\) | \(e\left(\frac{89}{239}\right)\) | \(e\left(\frac{130}{239}\right)\) | \(e\left(\frac{152}{239}\right)\) | \(e\left(\frac{15}{239}\right)\) | \(e\left(\frac{11}{239}\right)\) | \(e\left(\frac{94}{239}\right)\) | \(e\left(\frac{135}{239}\right)\) |
| \(\chi_{5269}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{158}{239}\right)\) | \(e\left(\frac{126}{239}\right)\) | \(e\left(\frac{77}{239}\right)\) | \(e\left(\frac{40}{239}\right)\) | \(e\left(\frac{45}{239}\right)\) | \(e\left(\frac{71}{239}\right)\) | \(e\left(\frac{235}{239}\right)\) | \(e\left(\frac{13}{239}\right)\) | \(e\left(\frac{198}{239}\right)\) | \(e\left(\frac{203}{239}\right)\) |
| \(\chi_{5269}(364,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{239}\right)\) | \(e\left(\frac{225}{239}\right)\) | \(e\left(\frac{18}{239}\right)\) | \(e\left(\frac{208}{239}\right)\) | \(e\left(\frac{234}{239}\right)\) | \(e\left(\frac{178}{239}\right)\) | \(e\left(\frac{27}{239}\right)\) | \(e\left(\frac{211}{239}\right)\) | \(e\left(\frac{217}{239}\right)\) | \(e\left(\frac{4}{239}\right)\) |
| \(\chi_{5269}(408,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{168}{239}\right)\) | \(e\left(\frac{137}{239}\right)\) | \(e\left(\frac{97}{239}\right)\) | \(e\left(\frac{218}{239}\right)\) | \(e\left(\frac{66}{239}\right)\) | \(e\left(\frac{136}{239}\right)\) | \(e\left(\frac{26}{239}\right)\) | \(e\left(\frac{35}{239}\right)\) | \(e\left(\frac{147}{239}\right)\) | \(e\left(\frac{234}{239}\right)\) |
| \(\chi_{5269}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{239}\right)\) | \(e\left(\frac{168}{239}\right)\) | \(e\left(\frac{23}{239}\right)\) | \(e\left(\frac{133}{239}\right)\) | \(e\left(\frac{60}{239}\right)\) | \(e\left(\frac{15}{239}\right)\) | \(e\left(\frac{154}{239}\right)\) | \(e\left(\frac{97}{239}\right)\) | \(e\left(\frac{25}{239}\right)\) | \(e\left(\frac{191}{239}\right)\) |
| \(\chi_{5269}(430,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{239}\right)\) | \(e\left(\frac{228}{239}\right)\) | \(e\left(\frac{219}{239}\right)\) | \(e\left(\frac{61}{239}\right)\) | \(e\left(\frac{218}{239}\right)\) | \(e\left(\frac{174}{239}\right)\) | \(e\left(\frac{209}{239}\right)\) | \(e\left(\frac{217}{239}\right)\) | \(e\left(\frac{51}{239}\right)\) | \(e\left(\frac{208}{239}\right)\) |
| \(\chi_{5269}(452,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{239}\right)\) | \(e\left(\frac{102}{239}\right)\) | \(e\left(\frac{142}{239}\right)\) | \(e\left(\frac{21}{239}\right)\) | \(e\left(\frac{173}{239}\right)\) | \(e\left(\frac{103}{239}\right)\) | \(e\left(\frac{213}{239}\right)\) | \(e\left(\frac{204}{239}\right)\) | \(e\left(\frac{92}{239}\right)\) | \(e\left(\frac{5}{239}\right)\) |
| \(\chi_{5269}(463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{144}{239}\right)\) | \(e\left(\frac{15}{239}\right)\) | \(e\left(\frac{49}{239}\right)\) | \(e\left(\frac{221}{239}\right)\) | \(e\left(\frac{159}{239}\right)\) | \(e\left(\frac{219}{239}\right)\) | \(e\left(\frac{193}{239}\right)\) | \(e\left(\frac{30}{239}\right)\) | \(e\left(\frac{126}{239}\right)\) | \(e\left(\frac{64}{239}\right)\) |
| \(\chi_{5269}(474,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{239}\right)\) | \(e\left(\frac{7}{239}\right)\) | \(e\left(\frac{230}{239}\right)\) | \(e\left(\frac{135}{239}\right)\) | \(e\left(\frac{122}{239}\right)\) | \(e\left(\frac{150}{239}\right)\) | \(e\left(\frac{106}{239}\right)\) | \(e\left(\frac{14}{239}\right)\) | \(e\left(\frac{11}{239}\right)\) | \(e\left(\frac{237}{239}\right)\) |
| \(\chi_{5269}(496,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{116}{239}\right)\) | \(e\left(\frac{32}{239}\right)\) | \(e\left(\frac{232}{239}\right)\) | \(e\left(\frac{105}{239}\right)\) | \(e\left(\frac{148}{239}\right)\) | \(e\left(\frac{37}{239}\right)\) | \(e\left(\frac{109}{239}\right)\) | \(e\left(\frac{64}{239}\right)\) | \(e\left(\frac{221}{239}\right)\) | \(e\left(\frac{25}{239}\right)\) |
| \(\chi_{5269}(518,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{239}\right)\) | \(e\left(\frac{18}{239}\right)\) | \(e\left(\frac{11}{239}\right)\) | \(e\left(\frac{74}{239}\right)\) | \(e\left(\frac{143}{239}\right)\) | \(e\left(\frac{215}{239}\right)\) | \(e\left(\frac{136}{239}\right)\) | \(e\left(\frac{36}{239}\right)\) | \(e\left(\frac{199}{239}\right)\) | \(e\left(\frac{29}{239}\right)\) |
| \(\chi_{5269}(562,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{239}\right)\) | \(e\left(\frac{135}{239}\right)\) | \(e\left(\frac{202}{239}\right)\) | \(e\left(\frac{77}{239}\right)\) | \(e\left(\frac{236}{239}\right)\) | \(e\left(\frac{59}{239}\right)\) | \(e\left(\frac{64}{239}\right)\) | \(e\left(\frac{31}{239}\right)\) | \(e\left(\frac{178}{239}\right)\) | \(e\left(\frac{98}{239}\right)\) |
| \(\chi_{5269}(573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{239}\right)\) | \(e\left(\frac{61}{239}\right)\) | \(e\left(\frac{24}{239}\right)\) | \(e\left(\frac{118}{239}\right)\) | \(e\left(\frac{73}{239}\right)\) | \(e\left(\frac{78}{239}\right)\) | \(e\left(\frac{36}{239}\right)\) | \(e\left(\frac{122}{239}\right)\) | \(e\left(\frac{130}{239}\right)\) | \(e\left(\frac{85}{239}\right)\) |
| \(\chi_{5269}(595,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{226}{239}\right)\) | \(e\left(\frac{153}{239}\right)\) | \(e\left(\frac{213}{239}\right)\) | \(e\left(\frac{151}{239}\right)\) | \(e\left(\frac{140}{239}\right)\) | \(e\left(\frac{35}{239}\right)\) | \(e\left(\frac{200}{239}\right)\) | \(e\left(\frac{67}{239}\right)\) | \(e\left(\frac{138}{239}\right)\) | \(e\left(\frac{127}{239}\right)\) |
| \(\chi_{5269}(606,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{82}{239}\right)\) | \(e\left(\frac{138}{239}\right)\) | \(e\left(\frac{164}{239}\right)\) | \(e\left(\frac{169}{239}\right)\) | \(e\left(\frac{220}{239}\right)\) | \(e\left(\frac{55}{239}\right)\) | \(e\left(\frac{7}{239}\right)\) | \(e\left(\frac{37}{239}\right)\) | \(e\left(\frac{12}{239}\right)\) | \(e\left(\frac{63}{239}\right)\) |
| \(\chi_{5269}(628,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{236}{239}\right)\) | \(e\left(\frac{164}{239}\right)\) | \(e\left(\frac{233}{239}\right)\) | \(e\left(\frac{90}{239}\right)\) | \(e\left(\frac{161}{239}\right)\) | \(e\left(\frac{100}{239}\right)\) | \(e\left(\frac{230}{239}\right)\) | \(e\left(\frac{89}{239}\right)\) | \(e\left(\frac{87}{239}\right)\) | \(e\left(\frac{158}{239}\right)\) |