from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1058, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1058))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1058\) | |
| Conductor: | \(529\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(253\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 529.g | 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: | $\Q(\zeta_{253})$ |
| Fixed field: | Number field defined by a degree 253 polynomial (not computed) |
First 31 of 220 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1058}(3,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{128}{253}\right)\) | \(e\left(\frac{8}{253}\right)\) | \(e\left(\frac{20}{253}\right)\) | \(e\left(\frac{3}{253}\right)\) | \(e\left(\frac{50}{253}\right)\) | \(e\left(\frac{2}{253}\right)\) | \(e\left(\frac{136}{253}\right)\) | \(e\left(\frac{232}{253}\right)\) | \(e\left(\frac{197}{253}\right)\) | \(e\left(\frac{148}{253}\right)\) |
| \(\chi_{1058}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{253}\right)\) | \(e\left(\frac{16}{253}\right)\) | \(e\left(\frac{40}{253}\right)\) | \(e\left(\frac{6}{253}\right)\) | \(e\left(\frac{100}{253}\right)\) | \(e\left(\frac{4}{253}\right)\) | \(e\left(\frac{19}{253}\right)\) | \(e\left(\frac{211}{253}\right)\) | \(e\left(\frac{141}{253}\right)\) | \(e\left(\frac{43}{253}\right)\) |
| \(\chi_{1058}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{253}\right)\) | \(e\left(\frac{95}{253}\right)\) | \(e\left(\frac{111}{253}\right)\) | \(e\left(\frac{4}{253}\right)\) | \(e\left(\frac{151}{253}\right)\) | \(e\left(\frac{87}{253}\right)\) | \(e\left(\frac{97}{253}\right)\) | \(e\left(\frac{225}{253}\right)\) | \(e\left(\frac{94}{253}\right)\) | \(e\left(\frac{113}{253}\right)\) |
| \(\chi_{1058}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{253}\right)\) | \(e\left(\frac{1}{253}\right)\) | \(e\left(\frac{129}{253}\right)\) | \(e\left(\frac{32}{253}\right)\) | \(e\left(\frac{196}{253}\right)\) | \(e\left(\frac{190}{253}\right)\) | \(e\left(\frac{17}{253}\right)\) | \(e\left(\frac{29}{253}\right)\) | \(e\left(\frac{246}{253}\right)\) | \(e\left(\frac{145}{253}\right)\) |
| \(\chi_{1058}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{253}\right)\) | \(e\left(\frac{24}{253}\right)\) | \(e\left(\frac{60}{253}\right)\) | \(e\left(\frac{9}{253}\right)\) | \(e\left(\frac{150}{253}\right)\) | \(e\left(\frac{6}{253}\right)\) | \(e\left(\frac{155}{253}\right)\) | \(e\left(\frac{190}{253}\right)\) | \(e\left(\frac{85}{253}\right)\) | \(e\left(\frac{191}{253}\right)\) |
| \(\chi_{1058}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{188}{253}\right)\) | \(e\left(\frac{75}{253}\right)\) | \(e\left(\frac{61}{253}\right)\) | \(e\left(\frac{123}{253}\right)\) | \(e\left(\frac{26}{253}\right)\) | \(e\left(\frac{82}{253}\right)\) | \(e\left(\frac{10}{253}\right)\) | \(e\left(\frac{151}{253}\right)\) | \(e\left(\frac{234}{253}\right)\) | \(e\left(\frac{249}{253}\right)\) |
| \(\chi_{1058}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{253}\right)\) | \(e\left(\frac{212}{253}\right)\) | \(e\left(\frac{24}{253}\right)\) | \(e\left(\frac{206}{253}\right)\) | \(e\left(\frac{60}{253}\right)\) | \(e\left(\frac{53}{253}\right)\) | \(e\left(\frac{62}{253}\right)\) | \(e\left(\frac{76}{253}\right)\) | \(e\left(\frac{34}{253}\right)\) | \(e\left(\frac{127}{253}\right)\) |
| \(\chi_{1058}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{253}\right)\) | \(e\left(\frac{65}{253}\right)\) | \(e\left(\frac{36}{253}\right)\) | \(e\left(\frac{56}{253}\right)\) | \(e\left(\frac{90}{253}\right)\) | \(e\left(\frac{206}{253}\right)\) | \(e\left(\frac{93}{253}\right)\) | \(e\left(\frac{114}{253}\right)\) | \(e\left(\frac{51}{253}\right)\) | \(e\left(\frac{64}{253}\right)\) |
| \(\chi_{1058}(39,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{130}{253}\right)\) | \(e\left(\frac{103}{253}\right)\) | \(e\left(\frac{131}{253}\right)\) | \(e\left(\frac{7}{253}\right)\) | \(e\left(\frac{201}{253}\right)\) | \(e\left(\frac{89}{253}\right)\) | \(e\left(\frac{233}{253}\right)\) | \(e\left(\frac{204}{253}\right)\) | \(e\left(\frac{38}{253}\right)\) | \(e\left(\frac{8}{253}\right)\) |
| \(\chi_{1058}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{162}{253}\right)\) | \(e\left(\frac{105}{253}\right)\) | \(e\left(\frac{136}{253}\right)\) | \(e\left(\frac{71}{253}\right)\) | \(e\left(\frac{87}{253}\right)\) | \(e\left(\frac{216}{253}\right)\) | \(e\left(\frac{14}{253}\right)\) | \(e\left(\frac{9}{253}\right)\) | \(e\left(\frac{24}{253}\right)\) | \(e\left(\frac{45}{253}\right)\) |
| \(\chi_{1058}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{253}\right)\) | \(e\left(\frac{129}{253}\right)\) | \(e\left(\frac{196}{253}\right)\) | \(e\left(\frac{80}{253}\right)\) | \(e\left(\frac{237}{253}\right)\) | \(e\left(\frac{222}{253}\right)\) | \(e\left(\frac{169}{253}\right)\) | \(e\left(\frac{199}{253}\right)\) | \(e\left(\frac{109}{253}\right)\) | \(e\left(\frac{236}{253}\right)\) |
| \(\chi_{1058}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{253}\right)\) | \(e\left(\frac{225}{253}\right)\) | \(e\left(\frac{183}{253}\right)\) | \(e\left(\frac{116}{253}\right)\) | \(e\left(\frac{78}{253}\right)\) | \(e\left(\frac{246}{253}\right)\) | \(e\left(\frac{30}{253}\right)\) | \(e\left(\frac{200}{253}\right)\) | \(e\left(\frac{196}{253}\right)\) | \(e\left(\frac{241}{253}\right)\) |
| \(\chi_{1058}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{253}\right)\) | \(e\left(\frac{84}{253}\right)\) | \(e\left(\frac{210}{253}\right)\) | \(e\left(\frac{158}{253}\right)\) | \(e\left(\frac{19}{253}\right)\) | \(e\left(\frac{21}{253}\right)\) | \(e\left(\frac{163}{253}\right)\) | \(e\left(\frac{159}{253}\right)\) | \(e\left(\frac{171}{253}\right)\) | \(e\left(\frac{36}{253}\right)\) |
| \(\chi_{1058}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{253}\right)\) | \(e\left(\frac{56}{253}\right)\) | \(e\left(\frac{140}{253}\right)\) | \(e\left(\frac{21}{253}\right)\) | \(e\left(\frac{97}{253}\right)\) | \(e\left(\frac{14}{253}\right)\) | \(e\left(\frac{193}{253}\right)\) | \(e\left(\frac{106}{253}\right)\) | \(e\left(\frac{114}{253}\right)\) | \(e\left(\frac{24}{253}\right)\) |
| \(\chi_{1058}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{253}\right)\) | \(e\left(\frac{178}{253}\right)\) | \(e\left(\frac{192}{253}\right)\) | \(e\left(\frac{130}{253}\right)\) | \(e\left(\frac{227}{253}\right)\) | \(e\left(\frac{171}{253}\right)\) | \(e\left(\frac{243}{253}\right)\) | \(e\left(\frac{102}{253}\right)\) | \(e\left(\frac{19}{253}\right)\) | \(e\left(\frac{4}{253}\right)\) |
| \(\chi_{1058}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{144}{253}\right)\) | \(e\left(\frac{9}{253}\right)\) | \(e\left(\frac{149}{253}\right)\) | \(e\left(\frac{35}{253}\right)\) | \(e\left(\frac{246}{253}\right)\) | \(e\left(\frac{192}{253}\right)\) | \(e\left(\frac{153}{253}\right)\) | \(e\left(\frac{8}{253}\right)\) | \(e\left(\frac{190}{253}\right)\) | \(e\left(\frac{40}{253}\right)\) |
| \(\chi_{1058}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{70}{253}\right)\) | \(e\left(\frac{36}{253}\right)\) | \(e\left(\frac{90}{253}\right)\) | \(e\left(\frac{140}{253}\right)\) | \(e\left(\frac{225}{253}\right)\) | \(e\left(\frac{9}{253}\right)\) | \(e\left(\frac{106}{253}\right)\) | \(e\left(\frac{32}{253}\right)\) | \(e\left(\frac{1}{253}\right)\) | \(e\left(\frac{160}{253}\right)\) |
| \(\chi_{1058}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{253}\right)\) | \(e\left(\frac{32}{253}\right)\) | \(e\left(\frac{80}{253}\right)\) | \(e\left(\frac{12}{253}\right)\) | \(e\left(\frac{200}{253}\right)\) | \(e\left(\frac{8}{253}\right)\) | \(e\left(\frac{38}{253}\right)\) | \(e\left(\frac{169}{253}\right)\) | \(e\left(\frac{29}{253}\right)\) | \(e\left(\frac{86}{253}\right)\) |
| \(\chi_{1058}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{240}{253}\right)\) | \(e\left(\frac{15}{253}\right)\) | \(e\left(\frac{164}{253}\right)\) | \(e\left(\frac{227}{253}\right)\) | \(e\left(\frac{157}{253}\right)\) | \(e\left(\frac{67}{253}\right)\) | \(e\left(\frac{2}{253}\right)\) | \(e\left(\frac{182}{253}\right)\) | \(e\left(\frac{148}{253}\right)\) | \(e\left(\frac{151}{253}\right)\) |
| \(\chi_{1058}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{253}\right)\) | \(e\left(\frac{83}{253}\right)\) | \(e\left(\frac{81}{253}\right)\) | \(e\left(\frac{126}{253}\right)\) | \(e\left(\frac{76}{253}\right)\) | \(e\left(\frac{84}{253}\right)\) | \(e\left(\frac{146}{253}\right)\) | \(e\left(\frac{130}{253}\right)\) | \(e\left(\frac{178}{253}\right)\) | \(e\left(\frac{144}{253}\right)\) |
| \(\chi_{1058}(95,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{253}\right)\) | \(e\left(\frac{250}{253}\right)\) | \(e\left(\frac{119}{253}\right)\) | \(e\left(\frac{157}{253}\right)\) | \(e\left(\frac{171}{253}\right)\) | \(e\left(\frac{189}{253}\right)\) | \(e\left(\frac{202}{253}\right)\) | \(e\left(\frac{166}{253}\right)\) | \(e\left(\frac{21}{253}\right)\) | \(e\left(\frac{71}{253}\right)\) |
| \(\chi_{1058}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{253}\right)\) | \(e\left(\frac{181}{253}\right)\) | \(e\left(\frac{73}{253}\right)\) | \(e\left(\frac{226}{253}\right)\) | \(e\left(\frac{56}{253}\right)\) | \(e\left(\frac{235}{253}\right)\) | \(e\left(\frac{41}{253}\right)\) | \(e\left(\frac{189}{253}\right)\) | \(e\left(\frac{251}{253}\right)\) | \(e\left(\frac{186}{253}\right)\) |
| \(\chi_{1058}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{156}{253}\right)\) | \(e\left(\frac{73}{253}\right)\) | \(e\left(\frac{56}{253}\right)\) | \(e\left(\frac{59}{253}\right)\) | \(e\left(\frac{140}{253}\right)\) | \(e\left(\frac{208}{253}\right)\) | \(e\left(\frac{229}{253}\right)\) | \(e\left(\frac{93}{253}\right)\) | \(e\left(\frac{248}{253}\right)\) | \(e\left(\frac{212}{253}\right)\) |
| \(\chi_{1058}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{253}\right)\) | \(e\left(\frac{111}{253}\right)\) | \(e\left(\frac{151}{253}\right)\) | \(e\left(\frac{10}{253}\right)\) | \(e\left(\frac{251}{253}\right)\) | \(e\left(\frac{91}{253}\right)\) | \(e\left(\frac{116}{253}\right)\) | \(e\left(\frac{183}{253}\right)\) | \(e\left(\frac{235}{253}\right)\) | \(e\left(\frac{156}{253}\right)\) |
| \(\chi_{1058}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{252}{253}\right)\) | \(e\left(\frac{79}{253}\right)\) | \(e\left(\frac{71}{253}\right)\) | \(e\left(\frac{251}{253}\right)\) | \(e\left(\frac{51}{253}\right)\) | \(e\left(\frac{83}{253}\right)\) | \(e\left(\frac{78}{253}\right)\) | \(e\left(\frac{14}{253}\right)\) | \(e\left(\frac{206}{253}\right)\) | \(e\left(\frac{70}{253}\right)\) |
| \(\chi_{1058}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{100}{253}\right)\) | \(e\left(\frac{196}{253}\right)\) | \(e\left(\frac{237}{253}\right)\) | \(e\left(\frac{200}{253}\right)\) | \(e\left(\frac{213}{253}\right)\) | \(e\left(\frac{49}{253}\right)\) | \(e\left(\frac{43}{253}\right)\) | \(e\left(\frac{118}{253}\right)\) | \(e\left(\frac{146}{253}\right)\) | \(e\left(\frac{84}{253}\right)\) |
| \(\chi_{1058}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{253}\right)\) | \(e\left(\frac{113}{253}\right)\) | \(e\left(\frac{156}{253}\right)\) | \(e\left(\frac{74}{253}\right)\) | \(e\left(\frac{137}{253}\right)\) | \(e\left(\frac{218}{253}\right)\) | \(e\left(\frac{150}{253}\right)\) | \(e\left(\frac{241}{253}\right)\) | \(e\left(\frac{221}{253}\right)\) | \(e\left(\frac{193}{253}\right)\) |
| \(\chi_{1058}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{237}{253}\right)\) | \(e\left(\frac{252}{253}\right)\) | \(e\left(\frac{124}{253}\right)\) | \(e\left(\frac{221}{253}\right)\) | \(e\left(\frac{57}{253}\right)\) | \(e\left(\frac{63}{253}\right)\) | \(e\left(\frac{236}{253}\right)\) | \(e\left(\frac{224}{253}\right)\) | \(e\left(\frac{7}{253}\right)\) | \(e\left(\frac{108}{253}\right)\) |
| \(\chi_{1058}(131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{253}\right)\) | \(e\left(\frac{180}{253}\right)\) | \(e\left(\frac{197}{253}\right)\) | \(e\left(\frac{194}{253}\right)\) | \(e\left(\frac{113}{253}\right)\) | \(e\left(\frac{45}{253}\right)\) | \(e\left(\frac{24}{253}\right)\) | \(e\left(\frac{160}{253}\right)\) | \(e\left(\frac{5}{253}\right)\) | \(e\left(\frac{41}{253}\right)\) |
| \(\chi_{1058}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{217}{253}\right)\) | \(e\left(\frac{61}{253}\right)\) | \(e\left(\frac{26}{253}\right)\) | \(e\left(\frac{181}{253}\right)\) | \(e\left(\frac{65}{253}\right)\) | \(e\left(\frac{205}{253}\right)\) | \(e\left(\frac{25}{253}\right)\) | \(e\left(\frac{251}{253}\right)\) | \(e\left(\frac{79}{253}\right)\) | \(e\left(\frac{243}{253}\right)\) |
| \(\chi_{1058}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{253}\right)\) | \(e\left(\frac{118}{253}\right)\) | \(e\left(\frac{42}{253}\right)\) | \(e\left(\frac{234}{253}\right)\) | \(e\left(\frac{105}{253}\right)\) | \(e\left(\frac{156}{253}\right)\) | \(e\left(\frac{235}{253}\right)\) | \(e\left(\frac{133}{253}\right)\) | \(e\left(\frac{186}{253}\right)\) | \(e\left(\frac{159}{253}\right)\) |