from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(862, base_ring=CyclotomicField(430))
M = H._module
chi = DirichletCharacter(H, M([342]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,862))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(862\) | |
| Conductor: | \(431\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(215\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 431.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: | $\Q(\zeta_{215})$ |
| Fixed field: | Number field defined by a degree 215 polynomial (not computed) |
First 31 of 168 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{862}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{2}{215}\right)\) | \(e\left(\frac{171}{215}\right)\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{98}{215}\right)\) | \(e\left(\frac{177}{215}\right)\) | \(e\left(\frac{152}{215}\right)\) | \(e\left(\frac{141}{215}\right)\) | \(e\left(\frac{84}{215}\right)\) | \(e\left(\frac{106}{215}\right)\) |
| \(\chi_{862}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{98}{215}\right)\) | \(e\left(\frac{209}{215}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{72}{215}\right)\) | \(e\left(\frac{73}{215}\right)\) | \(e\left(\frac{138}{215}\right)\) | \(e\left(\frac{29}{215}\right)\) | \(e\left(\frac{31}{215}\right)\) | \(e\left(\frac{34}{215}\right)\) |
| \(\chi_{862}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{152}{215}\right)\) | \(e\left(\frac{96}{215}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{138}{215}\right)\) | \(e\left(\frac{122}{215}\right)\) | \(e\left(\frac{157}{215}\right)\) | \(e\left(\frac{181}{215}\right)\) | \(e\left(\frac{149}{215}\right)\) | \(e\left(\frac{101}{215}\right)\) |
| \(\chi_{862}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{84}{215}\right)\) | \(e\left(\frac{87}{215}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{31}{215}\right)\) | \(e\left(\frac{124}{215}\right)\) | \(e\left(\frac{149}{215}\right)\) | \(e\left(\frac{117}{215}\right)\) | \(e\left(\frac{88}{215}\right)\) | \(e\left(\frac{152}{215}\right)\) |
| \(\chi_{862}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{43}\right)\) | \(e\left(\frac{61}{215}\right)\) | \(e\left(\frac{163}{215}\right)\) | \(e\left(\frac{24}{43}\right)\) | \(e\left(\frac{194}{215}\right)\) | \(e\left(\frac{131}{215}\right)\) | \(e\left(\frac{121}{215}\right)\) | \(e\left(\frac{108}{215}\right)\) | \(e\left(\frac{197}{215}\right)\) | \(e\left(\frac{8}{215}\right)\) |
| \(\chi_{862}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{4}{215}\right)\) | \(e\left(\frac{127}{215}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{196}{215}\right)\) | \(e\left(\frac{139}{215}\right)\) | \(e\left(\frac{89}{215}\right)\) | \(e\left(\frac{67}{215}\right)\) | \(e\left(\frac{168}{215}\right)\) | \(e\left(\frac{212}{215}\right)\) |
| \(\chi_{862}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{43}\right)\) | \(e\left(\frac{53}{215}\right)\) | \(e\left(\frac{124}{215}\right)\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{17}{215}\right)\) | \(e\left(\frac{68}{215}\right)\) | \(e\left(\frac{158}{215}\right)\) | \(e\left(\frac{189}{215}\right)\) | \(e\left(\frac{76}{215}\right)\) | \(e\left(\frac{14}{215}\right)\) |
| \(\chi_{862}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{33}{215}\right)\) | \(e\left(\frac{134}{215}\right)\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{112}{215}\right)\) | \(e\left(\frac{18}{215}\right)\) | \(e\left(\frac{143}{215}\right)\) | \(e\left(\frac{69}{215}\right)\) | \(e\left(\frac{96}{215}\right)\) | \(e\left(\frac{29}{215}\right)\) |
| \(\chi_{862}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{43}\right)\) | \(e\left(\frac{109}{215}\right)\) | \(e\left(\frac{182}{215}\right)\) | \(e\left(\frac{2}{43}\right)\) | \(e\left(\frac{181}{215}\right)\) | \(e\left(\frac{79}{215}\right)\) | \(e\left(\frac{114}{215}\right)\) | \(e\left(\frac{52}{215}\right)\) | \(e\left(\frac{63}{215}\right)\) | \(e\left(\frac{187}{215}\right)\) |
| \(\chi_{862}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{87}{215}\right)\) | \(e\left(\frac{21}{215}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{178}{215}\right)\) | \(e\left(\frac{67}{215}\right)\) | \(e\left(\frac{162}{215}\right)\) | \(e\left(\frac{6}{215}\right)\) | \(e\left(\frac{214}{215}\right)\) | \(e\left(\frac{96}{215}\right)\) |
| \(\chi_{862}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{43}\right)\) | \(e\left(\frac{127}{215}\right)\) | \(e\left(\frac{1}{215}\right)\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{203}{215}\right)\) | \(e\left(\frac{167}{215}\right)\) | \(e\left(\frac{192}{215}\right)\) | \(e\left(\frac{31}{215}\right)\) | \(e\left(\frac{174}{215}\right)\) | \(e\left(\frac{66}{215}\right)\) |
| \(\chi_{862}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{43}\right)\) | \(e\left(\frac{186}{215}\right)\) | \(e\left(\frac{208}{215}\right)\) | \(e\left(\frac{33}{43}\right)\) | \(e\left(\frac{84}{215}\right)\) | \(e\left(\frac{121}{215}\right)\) | \(e\left(\frac{161}{215}\right)\) | \(e\left(\frac{213}{215}\right)\) | \(e\left(\frac{72}{215}\right)\) | \(e\left(\frac{183}{215}\right)\) |
| \(\chi_{862}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{19}{215}\right)\) | \(e\left(\frac{12}{215}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{71}{215}\right)\) | \(e\left(\frac{69}{215}\right)\) | \(e\left(\frac{154}{215}\right)\) | \(e\left(\frac{157}{215}\right)\) | \(e\left(\frac{153}{215}\right)\) | \(e\left(\frac{147}{215}\right)\) |
| \(\chi_{862}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{124}{215}\right)\) | \(e\left(\frac{67}{215}\right)\) | \(e\left(\frac{22}{43}\right)\) | \(e\left(\frac{56}{215}\right)\) | \(e\left(\frac{9}{215}\right)\) | \(e\left(\frac{179}{215}\right)\) | \(e\left(\frac{142}{215}\right)\) | \(e\left(\frac{48}{215}\right)\) | \(e\left(\frac{122}{215}\right)\) |
| \(\chi_{862}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{43}\right)\) | \(e\left(\frac{34}{215}\right)\) | \(e\left(\frac{112}{215}\right)\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{161}{215}\right)\) | \(e\left(\frac{214}{215}\right)\) | \(e\left(\frac{4}{215}\right)\) | \(e\left(\frac{32}{215}\right)\) | \(e\left(\frac{138}{215}\right)\) | \(e\left(\frac{82}{215}\right)\) |
| \(\chi_{862}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{43}\right)\) | \(e\left(\frac{211}{215}\right)\) | \(e\left(\frac{88}{215}\right)\) | \(e\left(\frac{9}{43}\right)\) | \(e\left(\frac{19}{215}\right)\) | \(e\left(\frac{76}{215}\right)\) | \(e\left(\frac{126}{215}\right)\) | \(e\left(\frac{148}{215}\right)\) | \(e\left(\frac{47}{215}\right)\) | \(e\left(\frac{3}{215}\right)\) |
| \(\chi_{862}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{43}\right)\) | \(e\left(\frac{154}{215}\right)\) | \(e\left(\frac{52}{215}\right)\) | \(e\left(\frac{19}{43}\right)\) | \(e\left(\frac{21}{215}\right)\) | \(e\left(\frac{84}{215}\right)\) | \(e\left(\frac{94}{215}\right)\) | \(e\left(\frac{107}{215}\right)\) | \(e\left(\frac{18}{215}\right)\) | \(e\left(\frac{207}{215}\right)\) |
| \(\chi_{862}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{43}\right)\) | \(e\left(\frac{203}{215}\right)\) | \(e\left(\frac{49}{215}\right)\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{57}{215}\right)\) | \(e\left(\frac{13}{215}\right)\) | \(e\left(\frac{163}{215}\right)\) | \(e\left(\frac{14}{215}\right)\) | \(e\left(\frac{141}{215}\right)\) | \(e\left(\frac{9}{215}\right)\) |
| \(\chi_{862}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{43}\right)\) | \(e\left(\frac{133}{215}\right)\) | \(e\left(\frac{84}{215}\right)\) | \(e\left(\frac{34}{43}\right)\) | \(e\left(\frac{67}{215}\right)\) | \(e\left(\frac{53}{215}\right)\) | \(e\left(\frac{3}{215}\right)\) | \(e\left(\frac{24}{215}\right)\) | \(e\left(\frac{211}{215}\right)\) | \(e\left(\frac{169}{215}\right)\) |
| \(\chi_{862}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{43}\right)\) | \(e\left(\frac{13}{215}\right)\) | \(e\left(\frac{144}{215}\right)\) | \(e\left(\frac{3}{43}\right)\) | \(e\left(\frac{207}{215}\right)\) | \(e\left(\frac{183}{215}\right)\) | \(e\left(\frac{128}{215}\right)\) | \(e\left(\frac{164}{215}\right)\) | \(e\left(\frac{116}{215}\right)\) | \(e\left(\frac{44}{215}\right)\) |
| \(\chi_{862}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{183}{215}\right)\) | \(e\left(\frac{59}{215}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{152}{215}\right)\) | \(e\left(\frac{178}{215}\right)\) | \(e\left(\frac{148}{215}\right)\) | \(e\left(\frac{109}{215}\right)\) | \(e\left(\frac{161}{215}\right)\) | \(e\left(\frac{24}{215}\right)\) |
| \(\chi_{862}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{12}{215}\right)\) | \(e\left(\frac{166}{215}\right)\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{158}{215}\right)\) | \(e\left(\frac{202}{215}\right)\) | \(e\left(\frac{52}{215}\right)\) | \(e\left(\frac{201}{215}\right)\) | \(e\left(\frac{74}{215}\right)\) | \(e\left(\frac{206}{215}\right)\) |
| \(\chi_{862}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{42}{43}\right)\) | \(e\left(\frac{63}{215}\right)\) | \(e\left(\frac{119}{215}\right)\) | \(e\left(\frac{41}{43}\right)\) | \(e\left(\frac{77}{215}\right)\) | \(e\left(\frac{93}{215}\right)\) | \(e\left(\frac{58}{215}\right)\) | \(e\left(\frac{34}{215}\right)\) | \(e\left(\frac{66}{215}\right)\) | \(e\left(\frac{114}{215}\right)\) |
| \(\chi_{862}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{43}\right)\) | \(e\left(\frac{97}{215}\right)\) | \(e\left(\frac{16}{215}\right)\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{23}{215}\right)\) | \(e\left(\frac{92}{215}\right)\) | \(e\left(\frac{62}{215}\right)\) | \(e\left(\frac{66}{215}\right)\) | \(e\left(\frac{204}{215}\right)\) | \(e\left(\frac{196}{215}\right)\) |
| \(\chi_{862}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{43}\right)\) | \(e\left(\frac{196}{215}\right)\) | \(e\left(\frac{203}{215}\right)\) | \(e\left(\frac{32}{43}\right)\) | \(e\left(\frac{144}{215}\right)\) | \(e\left(\frac{146}{215}\right)\) | \(e\left(\frac{61}{215}\right)\) | \(e\left(\frac{58}{215}\right)\) | \(e\left(\frac{62}{215}\right)\) | \(e\left(\frac{68}{215}\right)\) |
| \(\chi_{862}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{44}{215}\right)\) | \(e\left(\frac{107}{215}\right)\) | \(e\left(\frac{30}{43}\right)\) | \(e\left(\frac{6}{215}\right)\) | \(e\left(\frac{24}{215}\right)\) | \(e\left(\frac{119}{215}\right)\) | \(e\left(\frac{92}{215}\right)\) | \(e\left(\frac{128}{215}\right)\) | \(e\left(\frac{182}{215}\right)\) |
| \(\chi_{862}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{43}\right)\) | \(e\left(\frac{6}{215}\right)\) | \(e\left(\frac{83}{215}\right)\) | \(e\left(\frac{8}{43}\right)\) | \(e\left(\frac{79}{215}\right)\) | \(e\left(\frac{101}{215}\right)\) | \(e\left(\frac{26}{215}\right)\) | \(e\left(\frac{208}{215}\right)\) | \(e\left(\frac{37}{215}\right)\) | \(e\left(\frac{103}{215}\right)\) |
| \(\chi_{862}(135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{43}\right)\) | \(e\left(\frac{22}{215}\right)\) | \(e\left(\frac{161}{215}\right)\) | \(e\left(\frac{15}{43}\right)\) | \(e\left(\frac{3}{215}\right)\) | \(e\left(\frac{12}{215}\right)\) | \(e\left(\frac{167}{215}\right)\) | \(e\left(\frac{46}{215}\right)\) | \(e\left(\frac{64}{215}\right)\) | \(e\left(\frac{91}{215}\right)\) |
| \(\chi_{862}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{43}\right)\) | \(e\left(\frac{16}{215}\right)\) | \(e\left(\frac{78}{215}\right)\) | \(e\left(\frac{7}{43}\right)\) | \(e\left(\frac{139}{215}\right)\) | \(e\left(\frac{126}{215}\right)\) | \(e\left(\frac{141}{215}\right)\) | \(e\left(\frac{53}{215}\right)\) | \(e\left(\frac{27}{215}\right)\) | \(e\left(\frac{203}{215}\right)\) |
| \(\chi_{862}(147,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{43}\right)\) | \(e\left(\frac{62}{215}\right)\) | \(e\left(\frac{141}{215}\right)\) | \(e\left(\frac{11}{43}\right)\) | \(e\left(\frac{28}{215}\right)\) | \(e\left(\frac{112}{215}\right)\) | \(e\left(\frac{197}{215}\right)\) | \(e\left(\frac{71}{215}\right)\) | \(e\left(\frac{24}{215}\right)\) | \(e\left(\frac{61}{215}\right)\) |
| \(\chi_{862}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{43}\right)\) | \(e\left(\frac{158}{215}\right)\) | \(e\left(\frac{179}{215}\right)\) | \(e\left(\frac{10}{43}\right)\) | \(e\left(\frac{2}{215}\right)\) | \(e\left(\frac{8}{215}\right)\) | \(e\left(\frac{183}{215}\right)\) | \(e\left(\frac{174}{215}\right)\) | \(e\left(\frac{186}{215}\right)\) | \(e\left(\frac{204}{215}\right)\) |