from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2012, base_ring=CyclotomicField(502))
M = H._module
chi = DirichletCharacter(H, M([0,312]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,2012))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2012\) | |
| Conductor: | \(503\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(251\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 503.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: | $\Q(\zeta_{251})$ |
| Fixed field: | Number field defined by a degree 251 polynomial (not computed) |
First 31 of 250 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2012}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{240}{251}\right)\) | \(e\left(\frac{156}{251}\right)\) | \(e\left(\frac{113}{251}\right)\) | \(e\left(\frac{229}{251}\right)\) | \(e\left(\frac{26}{251}\right)\) | \(e\left(\frac{17}{251}\right)\) | \(e\left(\frac{145}{251}\right)\) | \(e\left(\frac{82}{251}\right)\) | \(e\left(\frac{75}{251}\right)\) | \(e\left(\frac{102}{251}\right)\) |
| \(\chi_{2012}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{134}{251}\right)\) | \(e\left(\frac{62}{251}\right)\) | \(e\left(\frac{61}{251}\right)\) | \(e\left(\frac{17}{251}\right)\) | \(e\left(\frac{94}{251}\right)\) | \(e\left(\frac{158}{251}\right)\) | \(e\left(\frac{196}{251}\right)\) | \(e\left(\frac{142}{251}\right)\) | \(e\left(\frac{136}{251}\right)\) | \(e\left(\frac{195}{251}\right)\) |
| \(\chi_{2012}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{251}\right)\) | \(e\left(\frac{121}{251}\right)\) | \(e\left(\frac{115}{251}\right)\) | \(e\left(\frac{102}{251}\right)\) | \(e\left(\frac{62}{251}\right)\) | \(e\left(\frac{195}{251}\right)\) | \(e\left(\frac{172}{251}\right)\) | \(e\left(\frac{99}{251}\right)\) | \(e\left(\frac{63}{251}\right)\) | \(e\left(\frac{166}{251}\right)\) |
| \(\chi_{2012}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{156}{251}\right)\) | \(e\left(\frac{1}{251}\right)\) | \(e\left(\frac{86}{251}\right)\) | \(e\left(\frac{61}{251}\right)\) | \(e\left(\frac{42}{251}\right)\) | \(e\left(\frac{124}{251}\right)\) | \(e\left(\frac{157}{251}\right)\) | \(e\left(\frac{229}{251}\right)\) | \(e\left(\frac{237}{251}\right)\) | \(e\left(\frac{242}{251}\right)\) |
| \(\chi_{2012}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{251}\right)\) | \(e\left(\frac{99}{251}\right)\) | \(e\left(\frac{231}{251}\right)\) | \(e\left(\frac{15}{251}\right)\) | \(e\left(\frac{142}{251}\right)\) | \(e\left(\frac{228}{251}\right)\) | \(e\left(\frac{232}{251}\right)\) | \(e\left(\frac{81}{251}\right)\) | \(e\left(\frac{120}{251}\right)\) | \(e\left(\frac{113}{251}\right)\) |
| \(\chi_{2012}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{251}\right)\) | \(e\left(\frac{86}{251}\right)\) | \(e\left(\frac{117}{251}\right)\) | \(e\left(\frac{226}{251}\right)\) | \(e\left(\frac{98}{251}\right)\) | \(e\left(\frac{122}{251}\right)\) | \(e\left(\frac{199}{251}\right)\) | \(e\left(\frac{116}{251}\right)\) | \(e\left(\frac{51}{251}\right)\) | \(e\left(\frac{230}{251}\right)\) |
| \(\chi_{2012}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{251}\right)\) | \(e\left(\frac{152}{251}\right)\) | \(e\left(\frac{20}{251}\right)\) | \(e\left(\frac{236}{251}\right)\) | \(e\left(\frac{109}{251}\right)\) | \(e\left(\frac{23}{251}\right)\) | \(e\left(\frac{19}{251}\right)\) | \(e\left(\frac{170}{251}\right)\) | \(e\left(\frac{131}{251}\right)\) | \(e\left(\frac{138}{251}\right)\) |
| \(\chi_{2012}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{251}\right)\) | \(e\left(\frac{34}{251}\right)\) | \(e\left(\frac{163}{251}\right)\) | \(e\left(\frac{66}{251}\right)\) | \(e\left(\frac{173}{251}\right)\) | \(e\left(\frac{200}{251}\right)\) | \(e\left(\frac{67}{251}\right)\) | \(e\left(\frac{5}{251}\right)\) | \(e\left(\frac{26}{251}\right)\) | \(e\left(\frac{196}{251}\right)\) |
| \(\chi_{2012}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{251}\right)\) | \(e\left(\frac{9}{251}\right)\) | \(e\left(\frac{21}{251}\right)\) | \(e\left(\frac{47}{251}\right)\) | \(e\left(\frac{127}{251}\right)\) | \(e\left(\frac{112}{251}\right)\) | \(e\left(\frac{158}{251}\right)\) | \(e\left(\frac{53}{251}\right)\) | \(e\left(\frac{125}{251}\right)\) | \(e\left(\frac{170}{251}\right)\) |
| \(\chi_{2012}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{195}{251}\right)\) | \(e\left(\frac{64}{251}\right)\) | \(e\left(\frac{233}{251}\right)\) | \(e\left(\frac{139}{251}\right)\) | \(e\left(\frac{178}{251}\right)\) | \(e\left(\frac{155}{251}\right)\) | \(e\left(\frac{8}{251}\right)\) | \(e\left(\frac{98}{251}\right)\) | \(e\left(\frac{108}{251}\right)\) | \(e\left(\frac{177}{251}\right)\) |
| \(\chi_{2012}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{251}\right)\) | \(e\left(\frac{61}{251}\right)\) | \(e\left(\frac{226}{251}\right)\) | \(e\left(\frac{207}{251}\right)\) | \(e\left(\frac{52}{251}\right)\) | \(e\left(\frac{34}{251}\right)\) | \(e\left(\frac{39}{251}\right)\) | \(e\left(\frac{164}{251}\right)\) | \(e\left(\frac{150}{251}\right)\) | \(e\left(\frac{204}{251}\right)\) |
| \(\chi_{2012}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{251}\right)\) | \(e\left(\frac{115}{251}\right)\) | \(e\left(\frac{101}{251}\right)\) | \(e\left(\frac{238}{251}\right)\) | \(e\left(\frac{61}{251}\right)\) | \(e\left(\frac{204}{251}\right)\) | \(e\left(\frac{234}{251}\right)\) | \(e\left(\frac{231}{251}\right)\) | \(e\left(\frac{147}{251}\right)\) | \(e\left(\frac{220}{251}\right)\) |
| \(\chi_{2012}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{105}{251}\right)\) | \(e\left(\frac{131}{251}\right)\) | \(e\left(\frac{222}{251}\right)\) | \(e\left(\frac{210}{251}\right)\) | \(e\left(\frac{231}{251}\right)\) | \(e\left(\frac{180}{251}\right)\) | \(e\left(\frac{236}{251}\right)\) | \(e\left(\frac{130}{251}\right)\) | \(e\left(\frac{174}{251}\right)\) | \(e\left(\frac{76}{251}\right)\) |
| \(\chi_{2012}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{251}\right)\) | \(e\left(\frac{116}{251}\right)\) | \(e\left(\frac{187}{251}\right)\) | \(e\left(\frac{48}{251}\right)\) | \(e\left(\frac{103}{251}\right)\) | \(e\left(\frac{77}{251}\right)\) | \(e\left(\frac{140}{251}\right)\) | \(e\left(\frac{209}{251}\right)\) | \(e\left(\frac{133}{251}\right)\) | \(e\left(\frac{211}{251}\right)\) |
| \(\chi_{2012}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{251}\right)\) | \(e\left(\frac{218}{251}\right)\) | \(e\left(\frac{174}{251}\right)\) | \(e\left(\frac{246}{251}\right)\) | \(e\left(\frac{120}{251}\right)\) | \(e\left(\frac{175}{251}\right)\) | \(e\left(\frac{90}{251}\right)\) | \(e\left(\frac{224}{251}\right)\) | \(e\left(\frac{211}{251}\right)\) | \(e\left(\frac{46}{251}\right)\) |
| \(\chi_{2012}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{251}\right)\) | \(e\left(\frac{42}{251}\right)\) | \(e\left(\frac{98}{251}\right)\) | \(e\left(\frac{52}{251}\right)\) | \(e\left(\frac{7}{251}\right)\) | \(e\left(\frac{188}{251}\right)\) | \(e\left(\frac{68}{251}\right)\) | \(e\left(\frac{80}{251}\right)\) | \(e\left(\frac{165}{251}\right)\) | \(e\left(\frac{124}{251}\right)\) |
| \(\chi_{2012}(129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{248}{251}\right)\) | \(e\left(\frac{111}{251}\right)\) | \(e\left(\frac{8}{251}\right)\) | \(e\left(\frac{245}{251}\right)\) | \(e\left(\frac{144}{251}\right)\) | \(e\left(\frac{210}{251}\right)\) | \(e\left(\frac{108}{251}\right)\) | \(e\left(\frac{68}{251}\right)\) | \(e\left(\frac{203}{251}\right)\) | \(e\left(\frac{5}{251}\right)\) |
| \(\chi_{2012}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{232}{251}\right)\) | \(e\left(\frac{201}{251}\right)\) | \(e\left(\frac{218}{251}\right)\) | \(e\left(\frac{213}{251}\right)\) | \(e\left(\frac{159}{251}\right)\) | \(e\left(\frac{75}{251}\right)\) | \(e\left(\frac{182}{251}\right)\) | \(e\left(\frac{96}{251}\right)\) | \(e\left(\frac{198}{251}\right)\) | \(e\left(\frac{199}{251}\right)\) |
| \(\chi_{2012}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{251}\right)\) | \(e\left(\frac{113}{251}\right)\) | \(e\left(\frac{180}{251}\right)\) | \(e\left(\frac{116}{251}\right)\) | \(e\left(\frac{228}{251}\right)\) | \(e\left(\frac{207}{251}\right)\) | \(e\left(\frac{171}{251}\right)\) | \(e\left(\frac{24}{251}\right)\) | \(e\left(\frac{175}{251}\right)\) | \(e\left(\frac{238}{251}\right)\) |
| \(\chi_{2012}(161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{251}\right)\) | \(e\left(\frac{250}{251}\right)\) | \(e\left(\frac{165}{251}\right)\) | \(e\left(\frac{190}{251}\right)\) | \(e\left(\frac{209}{251}\right)\) | \(e\left(\frac{127}{251}\right)\) | \(e\left(\frac{94}{251}\right)\) | \(e\left(\frac{22}{251}\right)\) | \(e\left(\frac{14}{251}\right)\) | \(e\left(\frac{9}{251}\right)\) |
| \(\chi_{2012}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{251}\right)\) | \(e\left(\frac{124}{251}\right)\) | \(e\left(\frac{122}{251}\right)\) | \(e\left(\frac{34}{251}\right)\) | \(e\left(\frac{188}{251}\right)\) | \(e\left(\frac{65}{251}\right)\) | \(e\left(\frac{141}{251}\right)\) | \(e\left(\frac{33}{251}\right)\) | \(e\left(\frac{21}{251}\right)\) | \(e\left(\frac{139}{251}\right)\) |
| \(\chi_{2012}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{154}{251}\right)\) | \(e\left(\frac{75}{251}\right)\) | \(e\left(\frac{175}{251}\right)\) | \(e\left(\frac{57}{251}\right)\) | \(e\left(\frac{138}{251}\right)\) | \(e\left(\frac{13}{251}\right)\) | \(e\left(\frac{229}{251}\right)\) | \(e\left(\frac{107}{251}\right)\) | \(e\left(\frac{205}{251}\right)\) | \(e\left(\frac{78}{251}\right)\) |
| \(\chi_{2012}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{251}\right)\) | \(e\left(\frac{167}{251}\right)\) | \(e\left(\frac{55}{251}\right)\) | \(e\left(\frac{147}{251}\right)\) | \(e\left(\frac{237}{251}\right)\) | \(e\left(\frac{126}{251}\right)\) | \(e\left(\frac{115}{251}\right)\) | \(e\left(\frac{91}{251}\right)\) | \(e\left(\frac{172}{251}\right)\) | \(e\left(\frac{3}{251}\right)\) |
| \(\chi_{2012}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{251}\right)\) | \(e\left(\frac{186}{251}\right)\) | \(e\left(\frac{183}{251}\right)\) | \(e\left(\frac{51}{251}\right)\) | \(e\left(\frac{31}{251}\right)\) | \(e\left(\frac{223}{251}\right)\) | \(e\left(\frac{86}{251}\right)\) | \(e\left(\frac{175}{251}\right)\) | \(e\left(\frac{157}{251}\right)\) | \(e\left(\frac{83}{251}\right)\) |
| \(\chi_{2012}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{251}\right)\) | \(e\left(\frac{26}{251}\right)\) | \(e\left(\frac{228}{251}\right)\) | \(e\left(\frac{80}{251}\right)\) | \(e\left(\frac{88}{251}\right)\) | \(e\left(\frac{212}{251}\right)\) | \(e\left(\frac{66}{251}\right)\) | \(e\left(\frac{181}{251}\right)\) | \(e\left(\frac{138}{251}\right)\) | \(e\left(\frac{17}{251}\right)\) |
| \(\chi_{2012}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{224}{251}\right)\) | \(e\left(\frac{246}{251}\right)\) | \(e\left(\frac{72}{251}\right)\) | \(e\left(\frac{197}{251}\right)\) | \(e\left(\frac{41}{251}\right)\) | \(e\left(\frac{133}{251}\right)\) | \(e\left(\frac{219}{251}\right)\) | \(e\left(\frac{110}{251}\right)\) | \(e\left(\frac{70}{251}\right)\) | \(e\left(\frac{45}{251}\right)\) |
| \(\chi_{2012}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{251}\right)\) | \(e\left(\frac{125}{251}\right)\) | \(e\left(\frac{208}{251}\right)\) | \(e\left(\frac{95}{251}\right)\) | \(e\left(\frac{230}{251}\right)\) | \(e\left(\frac{189}{251}\right)\) | \(e\left(\frac{47}{251}\right)\) | \(e\left(\frac{11}{251}\right)\) | \(e\left(\frac{7}{251}\right)\) | \(e\left(\frac{130}{251}\right)\) |
| \(\chi_{2012}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{251}\right)\) | \(e\left(\frac{118}{251}\right)\) | \(e\left(\frac{108}{251}\right)\) | \(e\left(\frac{170}{251}\right)\) | \(e\left(\frac{187}{251}\right)\) | \(e\left(\frac{74}{251}\right)\) | \(e\left(\frac{203}{251}\right)\) | \(e\left(\frac{165}{251}\right)\) | \(e\left(\frac{105}{251}\right)\) | \(e\left(\frac{193}{251}\right)\) |
| \(\chi_{2012}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{251}\right)\) | \(e\left(\frac{157}{251}\right)\) | \(e\left(\frac{199}{251}\right)\) | \(e\left(\frac{39}{251}\right)\) | \(e\left(\frac{68}{251}\right)\) | \(e\left(\frac{141}{251}\right)\) | \(e\left(\frac{51}{251}\right)\) | \(e\left(\frac{60}{251}\right)\) | \(e\left(\frac{61}{251}\right)\) | \(e\left(\frac{93}{251}\right)\) |
| \(\chi_{2012}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{251}\right)\) | \(e\left(\frac{181}{251}\right)\) | \(e\left(\frac{4}{251}\right)\) | \(e\left(\frac{248}{251}\right)\) | \(e\left(\frac{72}{251}\right)\) | \(e\left(\frac{105}{251}\right)\) | \(e\left(\frac{54}{251}\right)\) | \(e\left(\frac{34}{251}\right)\) | \(e\left(\frac{227}{251}\right)\) | \(e\left(\frac{128}{251}\right)\) |
| \(\chi_{2012}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{130}{251}\right)\) | \(e\left(\frac{210}{251}\right)\) | \(e\left(\frac{239}{251}\right)\) | \(e\left(\frac{9}{251}\right)\) | \(e\left(\frac{35}{251}\right)\) | \(e\left(\frac{187}{251}\right)\) | \(e\left(\frac{89}{251}\right)\) | \(e\left(\frac{149}{251}\right)\) | \(e\left(\frac{72}{251}\right)\) | \(e\left(\frac{118}{251}\right)\) |