from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5269, base_ring=CyclotomicField(478))
M = H._module
chi = DirichletCharacter(H, M([0,214]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,5269))
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: | \(239\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 479.c | 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: | $\Q(\zeta_{239})$ |
| Fixed field: | Number field defined by a degree 239 polynomial (not computed) |
First 31 of 238 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5269}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{239}\right)\) | \(e\left(\frac{161}{239}\right)\) | \(e\left(\frac{32}{239}\right)\) | \(e\left(\frac{237}{239}\right)\) | \(e\left(\frac{177}{239}\right)\) | \(e\left(\frac{104}{239}\right)\) | \(e\left(\frac{48}{239}\right)\) | \(e\left(\frac{83}{239}\right)\) | \(e\left(\frac{14}{239}\right)\) | \(e\left(\frac{193}{239}\right)\) |
| \(\chi_{5269}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{239}\right)\) | \(e\left(\frac{218}{239}\right)\) | \(e\left(\frac{27}{239}\right)\) | \(e\left(\frac{73}{239}\right)\) | \(e\left(\frac{112}{239}\right)\) | \(e\left(\frac{28}{239}\right)\) | \(e\left(\frac{160}{239}\right)\) | \(e\left(\frac{197}{239}\right)\) | \(e\left(\frac{206}{239}\right)\) | \(e\left(\frac{6}{239}\right)\) |
| \(\chi_{5269}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{239}\right)\) | \(e\left(\frac{75}{239}\right)\) | \(e\left(\frac{6}{239}\right)\) | \(e\left(\frac{149}{239}\right)\) | \(e\left(\frac{78}{239}\right)\) | \(e\left(\frac{139}{239}\right)\) | \(e\left(\frac{9}{239}\right)\) | \(e\left(\frac{150}{239}\right)\) | \(e\left(\frac{152}{239}\right)\) | \(e\left(\frac{81}{239}\right)\) |
| \(\chi_{5269}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{239}\right)\) | \(e\left(\frac{185}{239}\right)\) | \(e\left(\frac{206}{239}\right)\) | \(e\left(\frac{17}{239}\right)\) | \(e\left(\frac{49}{239}\right)\) | \(e\left(\frac{72}{239}\right)\) | \(e\left(\frac{70}{239}\right)\) | \(e\left(\frac{131}{239}\right)\) | \(e\left(\frac{120}{239}\right)\) | \(e\left(\frac{152}{239}\right)\) |
| \(\chi_{5269}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{239}\right)\) | \(e\left(\frac{208}{239}\right)\) | \(e\left(\frac{74}{239}\right)\) | \(e\left(\frac{85}{239}\right)\) | \(e\left(\frac{6}{239}\right)\) | \(e\left(\frac{121}{239}\right)\) | \(e\left(\frac{111}{239}\right)\) | \(e\left(\frac{177}{239}\right)\) | \(e\left(\frac{122}{239}\right)\) | \(e\left(\frac{43}{239}\right)\) |
| \(\chi_{5269}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{239}\right)\) | \(e\left(\frac{141}{239}\right)\) | \(e\left(\frac{126}{239}\right)\) | \(e\left(\frac{22}{239}\right)\) | \(e\left(\frac{204}{239}\right)\) | \(e\left(\frac{51}{239}\right)\) | \(e\left(\frac{189}{239}\right)\) | \(e\left(\frac{43}{239}\right)\) | \(e\left(\frac{85}{239}\right)\) | \(e\left(\frac{28}{239}\right)\) |
| \(\chi_{5269}(122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{239}\right)\) | \(e\left(\frac{180}{239}\right)\) | \(e\left(\frac{110}{239}\right)\) | \(e\left(\frac{23}{239}\right)\) | \(e\left(\frac{235}{239}\right)\) | \(e\left(\frac{238}{239}\right)\) | \(e\left(\frac{165}{239}\right)\) | \(e\left(\frac{121}{239}\right)\) | \(e\left(\frac{78}{239}\right)\) | \(e\left(\frac{51}{239}\right)\) |
| \(\chi_{5269}(144,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{239}\right)\) | \(e\left(\frac{83}{239}\right)\) | \(e\left(\frac{64}{239}\right)\) | \(e\left(\frac{235}{239}\right)\) | \(e\left(\frac{115}{239}\right)\) | \(e\left(\frac{208}{239}\right)\) | \(e\left(\frac{96}{239}\right)\) | \(e\left(\frac{166}{239}\right)\) | \(e\left(\frac{28}{239}\right)\) | \(e\left(\frac{147}{239}\right)\) |
| \(\chi_{5269}(210,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{90}{239}\right)\) | \(e\left(\frac{99}{239}\right)\) | \(e\left(\frac{180}{239}\right)\) | \(e\left(\frac{168}{239}\right)\) | \(e\left(\frac{189}{239}\right)\) | \(e\left(\frac{107}{239}\right)\) | \(e\left(\frac{31}{239}\right)\) | \(e\left(\frac{198}{239}\right)\) | \(e\left(\frac{19}{239}\right)\) | \(e\left(\frac{40}{239}\right)\) |
| \(\chi_{5269}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{239}\right)\) | \(e\left(\frac{16}{239}\right)\) | \(e\left(\frac{116}{239}\right)\) | \(e\left(\frac{172}{239}\right)\) | \(e\left(\frac{74}{239}\right)\) | \(e\left(\frac{138}{239}\right)\) | \(e\left(\frac{174}{239}\right)\) | \(e\left(\frac{32}{239}\right)\) | \(e\left(\frac{230}{239}\right)\) | \(e\left(\frac{132}{239}\right)\) |
| \(\chi_{5269}(243,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{198}{239}\right)\) | \(e\left(\frac{170}{239}\right)\) | \(e\left(\frac{157}{239}\right)\) | \(e\left(\frac{35}{239}\right)\) | \(e\left(\frac{129}{239}\right)\) | \(e\left(\frac{92}{239}\right)\) | \(e\left(\frac{116}{239}\right)\) | \(e\left(\frac{101}{239}\right)\) | \(e\left(\frac{233}{239}\right)\) | \(e\left(\frac{88}{239}\right)\) |
| \(\chi_{5269}(276,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{239}\right)\) | \(e\left(\frac{140}{239}\right)\) | \(e\left(\frac{59}{239}\right)\) | \(e\left(\frac{71}{239}\right)\) | \(e\left(\frac{50}{239}\right)\) | \(e\left(\frac{132}{239}\right)\) | \(e\left(\frac{208}{239}\right)\) | \(e\left(\frac{41}{239}\right)\) | \(e\left(\frac{220}{239}\right)\) | \(e\left(\frac{199}{239}\right)\) |
| \(\chi_{5269}(309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{239}\right)\) | \(e\left(\frac{222}{239}\right)\) | \(e\left(\frac{56}{239}\right)\) | \(e\left(\frac{116}{239}\right)\) | \(e\left(\frac{11}{239}\right)\) | \(e\left(\frac{182}{239}\right)\) | \(e\left(\frac{84}{239}\right)\) | \(e\left(\frac{205}{239}\right)\) | \(e\left(\frac{144}{239}\right)\) | \(e\left(\frac{39}{239}\right)\) |
| \(\chi_{5269}(320,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{239}\right)\) | \(e\left(\frac{149}{239}\right)\) | \(e\left(\frac{184}{239}\right)\) | \(e\left(\frac{108}{239}\right)\) | \(e\left(\frac{2}{239}\right)\) | \(e\left(\frac{120}{239}\right)\) | \(e\left(\frac{37}{239}\right)\) | \(e\left(\frac{59}{239}\right)\) | \(e\left(\frac{200}{239}\right)\) | \(e\left(\frac{94}{239}\right)\) |
| \(\chi_{5269}(331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{166}{239}\right)\) | \(e\left(\frac{87}{239}\right)\) | \(e\left(\frac{93}{239}\right)\) | \(e\left(\frac{39}{239}\right)\) | \(e\left(\frac{14}{239}\right)\) | \(e\left(\frac{123}{239}\right)\) | \(e\left(\frac{20}{239}\right)\) | \(e\left(\frac{174}{239}\right)\) | \(e\left(\frac{205}{239}\right)\) | \(e\left(\frac{180}{239}\right)\) |
| \(\chi_{5269}(375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{239}\right)\) | \(e\left(\frac{55}{239}\right)\) | \(e\left(\frac{100}{239}\right)\) | \(e\left(\frac{173}{239}\right)\) | \(e\left(\frac{105}{239}\right)\) | \(e\left(\frac{86}{239}\right)\) | \(e\left(\frac{150}{239}\right)\) | \(e\left(\frac{110}{239}\right)\) | \(e\left(\frac{223}{239}\right)\) | \(e\left(\frac{155}{239}\right)\) |
| \(\chi_{5269}(386,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{155}{239}\right)\) | \(e\left(\frac{51}{239}\right)\) | \(e\left(\frac{71}{239}\right)\) | \(e\left(\frac{130}{239}\right)\) | \(e\left(\frac{206}{239}\right)\) | \(e\left(\frac{171}{239}\right)\) | \(e\left(\frac{226}{239}\right)\) | \(e\left(\frac{102}{239}\right)\) | \(e\left(\frac{46}{239}\right)\) | \(e\left(\frac{122}{239}\right)\) |
| \(\chi_{5269}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{239}\right)\) | \(e\left(\frac{6}{239}\right)\) | \(e\left(\frac{163}{239}\right)\) | \(e\left(\frac{184}{239}\right)\) | \(e\left(\frac{207}{239}\right)\) | \(e\left(\frac{231}{239}\right)\) | \(e\left(\frac{125}{239}\right)\) | \(e\left(\frac{12}{239}\right)\) | \(e\left(\frac{146}{239}\right)\) | \(e\left(\frac{169}{239}\right)\) |
| \(\chi_{5269}(441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{239}\right)\) | \(e\left(\frac{57}{239}\right)\) | \(e\left(\frac{234}{239}\right)\) | \(e\left(\frac{75}{239}\right)\) | \(e\left(\frac{174}{239}\right)\) | \(e\left(\frac{163}{239}\right)\) | \(e\left(\frac{112}{239}\right)\) | \(e\left(\frac{114}{239}\right)\) | \(e\left(\frac{192}{239}\right)\) | \(e\left(\frac{52}{239}\right)\) |
| \(\chi_{5269}(485,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{219}{239}\right)\) | \(e\left(\frac{217}{239}\right)\) | \(e\left(\frac{199}{239}\right)\) | \(e\left(\frac{122}{239}\right)\) | \(e\left(\frac{197}{239}\right)\) | \(e\left(\frac{109}{239}\right)\) | \(e\left(\frac{179}{239}\right)\) | \(e\left(\frac{195}{239}\right)\) | \(e\left(\frac{102}{239}\right)\) | \(e\left(\frac{177}{239}\right)\) |
| \(\chi_{5269}(507,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{239}\right)\) | \(e\left(\frac{2}{239}\right)\) | \(e\left(\frac{134}{239}\right)\) | \(e\left(\frac{141}{239}\right)\) | \(e\left(\frac{69}{239}\right)\) | \(e\left(\frac{77}{239}\right)\) | \(e\left(\frac{201}{239}\right)\) | \(e\left(\frac{4}{239}\right)\) | \(e\left(\frac{208}{239}\right)\) | \(e\left(\frac{136}{239}\right)\) |
| \(\chi_{5269}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{239}\right)\) | \(e\left(\frac{197}{239}\right)\) | \(e\left(\frac{54}{239}\right)\) | \(e\left(\frac{146}{239}\right)\) | \(e\left(\frac{224}{239}\right)\) | \(e\left(\frac{56}{239}\right)\) | \(e\left(\frac{81}{239}\right)\) | \(e\left(\frac{155}{239}\right)\) | \(e\left(\frac{173}{239}\right)\) | \(e\left(\frac{12}{239}\right)\) |
| \(\chi_{5269}(540,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{239}\right)\) | \(e\left(\frac{236}{239}\right)\) | \(e\left(\frac{38}{239}\right)\) | \(e\left(\frac{147}{239}\right)\) | \(e\left(\frac{16}{239}\right)\) | \(e\left(\frac{4}{239}\right)\) | \(e\left(\frac{57}{239}\right)\) | \(e\left(\frac{233}{239}\right)\) | \(e\left(\frac{166}{239}\right)\) | \(e\left(\frac{35}{239}\right)\) |
| \(\chi_{5269}(551,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{235}{239}\right)\) | \(e\left(\frac{139}{239}\right)\) | \(e\left(\frac{231}{239}\right)\) | \(e\left(\frac{120}{239}\right)\) | \(e\left(\frac{135}{239}\right)\) | \(e\left(\frac{213}{239}\right)\) | \(e\left(\frac{227}{239}\right)\) | \(e\left(\frac{39}{239}\right)\) | \(e\left(\frac{116}{239}\right)\) | \(e\left(\frac{131}{239}\right)\) |
| \(\chi_{5269}(584,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{239}\right)\) | \(e\left(\frac{155}{239}\right)\) | \(e\left(\frac{108}{239}\right)\) | \(e\left(\frac{53}{239}\right)\) | \(e\left(\frac{209}{239}\right)\) | \(e\left(\frac{112}{239}\right)\) | \(e\left(\frac{162}{239}\right)\) | \(e\left(\frac{71}{239}\right)\) | \(e\left(\frac{107}{239}\right)\) | \(e\left(\frac{24}{239}\right)\) |
| \(\chi_{5269}(617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{239}\right)\) | \(e\left(\frac{196}{239}\right)\) | \(e\left(\frac{226}{239}\right)\) | \(e\left(\frac{195}{239}\right)\) | \(e\left(\frac{70}{239}\right)\) | \(e\left(\frac{137}{239}\right)\) | \(e\left(\frac{100}{239}\right)\) | \(e\left(\frac{153}{239}\right)\) | \(e\left(\frac{69}{239}\right)\) | \(e\left(\frac{183}{239}\right)\) |
| \(\chi_{5269}(639,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{239}\right)\) | \(e\left(\frac{205}{239}\right)\) | \(e\left(\frac{112}{239}\right)\) | \(e\left(\frac{232}{239}\right)\) | \(e\left(\frac{22}{239}\right)\) | \(e\left(\frac{125}{239}\right)\) | \(e\left(\frac{168}{239}\right)\) | \(e\left(\frac{171}{239}\right)\) | \(e\left(\frac{49}{239}\right)\) | \(e\left(\frac{78}{239}\right)\) |
| \(\chi_{5269}(672,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{239}\right)\) | \(e\left(\frac{107}{239}\right)\) | \(e\left(\frac{238}{239}\right)\) | \(e\left(\frac{15}{239}\right)\) | \(e\left(\frac{226}{239}\right)\) | \(e\left(\frac{176}{239}\right)\) | \(e\left(\frac{118}{239}\right)\) | \(e\left(\frac{214}{239}\right)\) | \(e\left(\frac{134}{239}\right)\) | \(e\left(\frac{106}{239}\right)\) |
| \(\chi_{5269}(749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{222}{239}\right)\) | \(e\left(\frac{53}{239}\right)\) | \(e\left(\frac{205}{239}\right)\) | \(e\left(\frac{32}{239}\right)\) | \(e\left(\frac{36}{239}\right)\) | \(e\left(\frac{9}{239}\right)\) | \(e\left(\frac{188}{239}\right)\) | \(e\left(\frac{106}{239}\right)\) | \(e\left(\frac{15}{239}\right)\) | \(e\left(\frac{19}{239}\right)\) |
| \(\chi_{5269}(771,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{239}\right)\) | \(e\left(\frac{211}{239}\right)\) | \(e\left(\frac{36}{239}\right)\) | \(e\left(\frac{177}{239}\right)\) | \(e\left(\frac{229}{239}\right)\) | \(e\left(\frac{117}{239}\right)\) | \(e\left(\frac{54}{239}\right)\) | \(e\left(\frac{183}{239}\right)\) | \(e\left(\frac{195}{239}\right)\) | \(e\left(\frac{8}{239}\right)\) |
| \(\chi_{5269}(815,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{239}\right)\) | \(e\left(\frac{163}{239}\right)\) | \(e\left(\frac{166}{239}\right)\) | \(e\left(\frac{139}{239}\right)\) | \(e\left(\frac{7}{239}\right)\) | \(e\left(\frac{181}{239}\right)\) | \(e\left(\frac{10}{239}\right)\) | \(e\left(\frac{87}{239}\right)\) | \(e\left(\frac{222}{239}\right)\) | \(e\left(\frac{90}{239}\right)\) |