from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1097, base_ring=CyclotomicField(1096))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,1097))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1097\) | |
| Conductor: | \(1097\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1096\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{1096})$ |
| Fixed field: | Number field defined by a degree 1096 polynomial (not computed) |
First 31 of 544 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1097}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{274}\right)\) | \(e\left(\frac{1}{1096}\right)\) | \(e\left(\frac{33}{137}\right)\) | \(e\left(\frac{1003}{1096}\right)\) | \(e\left(\frac{133}{1096}\right)\) | \(e\left(\frac{555}{1096}\right)\) | \(e\left(\frac{99}{274}\right)\) | \(e\left(\frac{1}{548}\right)\) | \(e\left(\frac{39}{1096}\right)\) | \(e\left(\frac{1019}{1096}\right)\) |
| \(\chi_{1097}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{219}{274}\right)\) | \(e\left(\frac{1003}{1096}\right)\) | \(e\left(\frac{82}{137}\right)\) | \(e\left(\frac{977}{1096}\right)\) | \(e\left(\frac{783}{1096}\right)\) | \(e\left(\frac{993}{1096}\right)\) | \(e\left(\frac{109}{274}\right)\) | \(e\left(\frac{455}{548}\right)\) | \(e\left(\frac{757}{1096}\right)\) | \(e\left(\frac{585}{1096}\right)\) |
| \(\chi_{1097}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{274}\right)\) | \(e\left(\frac{133}{1096}\right)\) | \(e\left(\frac{5}{137}\right)\) | \(e\left(\frac{783}{1096}\right)\) | \(e\left(\frac{153}{1096}\right)\) | \(e\left(\frac{383}{1096}\right)\) | \(e\left(\frac{15}{274}\right)\) | \(e\left(\frac{133}{548}\right)\) | \(e\left(\frac{803}{1096}\right)\) | \(e\left(\frac{719}{1096}\right)\) |
| \(\chi_{1097}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{231}{274}\right)\) | \(e\left(\frac{555}{1096}\right)\) | \(e\left(\frac{94}{137}\right)\) | \(e\left(\frac{993}{1096}\right)\) | \(e\left(\frac{383}{1096}\right)\) | \(e\left(\frac{49}{1096}\right)\) | \(e\left(\frac{145}{274}\right)\) | \(e\left(\frac{7}{548}\right)\) | \(e\left(\frac{821}{1096}\right)\) | \(e\left(\frac{9}{1096}\right)\) |
| \(\chi_{1097}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{274}\right)\) | \(e\left(\frac{39}{1096}\right)\) | \(e\left(\frac{54}{137}\right)\) | \(e\left(\frac{757}{1096}\right)\) | \(e\left(\frac{803}{1096}\right)\) | \(e\left(\frac{821}{1096}\right)\) | \(e\left(\frac{25}{274}\right)\) | \(e\left(\frac{39}{548}\right)\) | \(e\left(\frac{425}{1096}\right)\) | \(e\left(\frac{285}{1096}\right)\) |
| \(\chi_{1097}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{274}\right)\) | \(e\left(\frac{1019}{1096}\right)\) | \(e\left(\frac{62}{137}\right)\) | \(e\left(\frac{585}{1096}\right)\) | \(e\left(\frac{719}{1096}\right)\) | \(e\left(\frac{9}{1096}\right)\) | \(e\left(\frac{49}{274}\right)\) | \(e\left(\frac{471}{548}\right)\) | \(e\left(\frac{285}{1096}\right)\) | \(e\left(\frac{449}{1096}\right)\) |
| \(\chi_{1097}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{274}\right)\) | \(e\left(\frac{265}{1096}\right)\) | \(e\left(\frac{114}{137}\right)\) | \(e\left(\frac{563}{1096}\right)\) | \(e\left(\frac{173}{1096}\right)\) | \(e\left(\frac{211}{1096}\right)\) | \(e\left(\frac{205}{274}\right)\) | \(e\left(\frac{265}{548}\right)\) | \(e\left(\frac{471}{1096}\right)\) | \(e\left(\frac{419}{1096}\right)\) |
| \(\chi_{1097}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{274}\right)\) | \(e\left(\frac{731}{1096}\right)\) | \(e\left(\frac{11}{137}\right)\) | \(e\left(\frac{1065}{1096}\right)\) | \(e\left(\frac{775}{1096}\right)\) | \(e\left(\frac{185}{1096}\right)\) | \(e\left(\frac{33}{274}\right)\) | \(e\left(\frac{183}{548}\right)\) | \(e\left(\frac{13}{1096}\right)\) | \(e\left(\frac{705}{1096}\right)\) |
| \(\chi_{1097}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{274}\right)\) | \(e\left(\frac{687}{1096}\right)\) | \(e\left(\frac{66}{137}\right)\) | \(e\left(\frac{773}{1096}\right)\) | \(e\left(\frac{403}{1096}\right)\) | \(e\left(\frac{973}{1096}\right)\) | \(e\left(\frac{61}{274}\right)\) | \(e\left(\frac{139}{548}\right)\) | \(e\left(\frac{489}{1096}\right)\) | \(e\left(\frac{805}{1096}\right)\) |
| \(\chi_{1097}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{274}\right)\) | \(e\left(\frac{617}{1096}\right)\) | \(e\left(\frac{85}{137}\right)\) | \(e\left(\frac{707}{1096}\right)\) | \(e\left(\frac{957}{1096}\right)\) | \(e\left(\frac{483}{1096}\right)\) | \(e\left(\frac{255}{274}\right)\) | \(e\left(\frac{69}{548}\right)\) | \(e\left(\frac{1047}{1096}\right)\) | \(e\left(\frac{715}{1096}\right)\) |
| \(\chi_{1097}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{274}\right)\) | \(e\left(\frac{171}{1096}\right)\) | \(e\left(\frac{26}{137}\right)\) | \(e\left(\frac{537}{1096}\right)\) | \(e\left(\frac{823}{1096}\right)\) | \(e\left(\frac{649}{1096}\right)\) | \(e\left(\frac{215}{274}\right)\) | \(e\left(\frac{171}{548}\right)\) | \(e\left(\frac{93}{1096}\right)\) | \(e\left(\frac{1081}{1096}\right)\) |
| \(\chi_{1097}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{274}\right)\) | \(e\left(\frac{55}{1096}\right)\) | \(e\left(\frac{34}{137}\right)\) | \(e\left(\frac{365}{1096}\right)\) | \(e\left(\frac{739}{1096}\right)\) | \(e\left(\frac{933}{1096}\right)\) | \(e\left(\frac{239}{274}\right)\) | \(e\left(\frac{55}{548}\right)\) | \(e\left(\frac{1049}{1096}\right)\) | \(e\left(\frac{149}{1096}\right)\) |
| \(\chi_{1097}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{274}\right)\) | \(e\left(\frac{397}{1096}\right)\) | \(e\left(\frac{86}{137}\right)\) | \(e\left(\frac{343}{1096}\right)\) | \(e\left(\frac{193}{1096}\right)\) | \(e\left(\frac{39}{1096}\right)\) | \(e\left(\frac{121}{274}\right)\) | \(e\left(\frac{397}{548}\right)\) | \(e\left(\frac{139}{1096}\right)\) | \(e\left(\frac{119}{1096}\right)\) |
| \(\chi_{1097}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{274}\right)\) | \(e\left(\frac{863}{1096}\right)\) | \(e\left(\frac{120}{137}\right)\) | \(e\left(\frac{845}{1096}\right)\) | \(e\left(\frac{795}{1096}\right)\) | \(e\left(\frac{13}{1096}\right)\) | \(e\left(\frac{223}{274}\right)\) | \(e\left(\frac{315}{548}\right)\) | \(e\left(\frac{777}{1096}\right)\) | \(e\left(\frac{405}{1096}\right)\) |
| \(\chi_{1097}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{274}\right)\) | \(e\left(\frac{3}{1096}\right)\) | \(e\left(\frac{99}{137}\right)\) | \(e\left(\frac{817}{1096}\right)\) | \(e\left(\frac{399}{1096}\right)\) | \(e\left(\frac{569}{1096}\right)\) | \(e\left(\frac{23}{274}\right)\) | \(e\left(\frac{3}{548}\right)\) | \(e\left(\frac{117}{1096}\right)\) | \(e\left(\frac{865}{1096}\right)\) |
| \(\chi_{1097}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{274}\right)\) | \(e\left(\frac{819}{1096}\right)\) | \(e\left(\frac{38}{137}\right)\) | \(e\left(\frac{553}{1096}\right)\) | \(e\left(\frac{423}{1096}\right)\) | \(e\left(\frac{801}{1096}\right)\) | \(e\left(\frac{251}{274}\right)\) | \(e\left(\frac{271}{548}\right)\) | \(e\left(\frac{157}{1096}\right)\) | \(e\left(\frac{505}{1096}\right)\) |
| \(\chi_{1097}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{75}{274}\right)\) | \(e\left(\frac{351}{1096}\right)\) | \(e\left(\frac{75}{137}\right)\) | \(e\left(\frac{237}{1096}\right)\) | \(e\left(\frac{651}{1096}\right)\) | \(e\left(\frac{813}{1096}\right)\) | \(e\left(\frac{225}{274}\right)\) | \(e\left(\frac{351}{548}\right)\) | \(e\left(\frac{537}{1096}\right)\) | \(e\left(\frac{373}{1096}\right)\) |
| \(\chi_{1097}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{274}\right)\) | \(e\left(\frac{691}{1096}\right)\) | \(e\left(\frac{61}{137}\right)\) | \(e\left(\frac{401}{1096}\right)\) | \(e\left(\frac{935}{1096}\right)\) | \(e\left(\frac{1001}{1096}\right)\) | \(e\left(\frac{183}{274}\right)\) | \(e\left(\frac{143}{548}\right)\) | \(e\left(\frac{645}{1096}\right)\) | \(e\left(\frac{497}{1096}\right)\) |
| \(\chi_{1097}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{274}\right)\) | \(e\left(\frac{749}{1096}\right)\) | \(e\left(\frac{57}{137}\right)\) | \(e\left(\frac{487}{1096}\right)\) | \(e\left(\frac{977}{1096}\right)\) | \(e\left(\frac{311}{1096}\right)\) | \(e\left(\frac{171}{274}\right)\) | \(e\left(\frac{201}{548}\right)\) | \(e\left(\frac{715}{1096}\right)\) | \(e\left(\frac{415}{1096}\right)\) |
| \(\chi_{1097}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{135}{274}\right)\) | \(e\left(\frac{303}{1096}\right)\) | \(e\left(\frac{135}{137}\right)\) | \(e\left(\frac{317}{1096}\right)\) | \(e\left(\frac{843}{1096}\right)\) | \(e\left(\frac{477}{1096}\right)\) | \(e\left(\frac{131}{274}\right)\) | \(e\left(\frac{303}{548}\right)\) | \(e\left(\frac{857}{1096}\right)\) | \(e\left(\frac{781}{1096}\right)\) |
| \(\chi_{1097}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{274}\right)\) | \(e\left(\frac{789}{1096}\right)\) | \(e\left(\frac{7}{137}\right)\) | \(e\left(\frac{55}{1096}\right)\) | \(e\left(\frac{817}{1096}\right)\) | \(e\left(\frac{591}{1096}\right)\) | \(e\left(\frac{21}{274}\right)\) | \(e\left(\frac{241}{548}\right)\) | \(e\left(\frac{83}{1096}\right)\) | \(e\left(\frac{623}{1096}\right)\) |
| \(\chi_{1097}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{274}\right)\) | \(e\left(\frac{187}{1096}\right)\) | \(e\left(\frac{6}{137}\right)\) | \(e\left(\frac{145}{1096}\right)\) | \(e\left(\frac{759}{1096}\right)\) | \(e\left(\frac{761}{1096}\right)\) | \(e\left(\frac{155}{274}\right)\) | \(e\left(\frac{187}{548}\right)\) | \(e\left(\frac{717}{1096}\right)\) | \(e\left(\frac{945}{1096}\right)\) |
| \(\chi_{1097}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{274}\right)\) | \(e\left(\frac{1005}{1096}\right)\) | \(e\left(\frac{11}{137}\right)\) | \(e\left(\frac{791}{1096}\right)\) | \(e\left(\frac{1049}{1096}\right)\) | \(e\left(\frac{1007}{1096}\right)\) | \(e\left(\frac{33}{274}\right)\) | \(e\left(\frac{457}{548}\right)\) | \(e\left(\frac{835}{1096}\right)\) | \(e\left(\frac{431}{1096}\right)\) |
| \(\chi_{1097}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{195}{274}\right)\) | \(e\left(\frac{529}{1096}\right)\) | \(e\left(\frac{58}{137}\right)\) | \(e\left(\frac{123}{1096}\right)\) | \(e\left(\frac{213}{1096}\right)\) | \(e\left(\frac{963}{1096}\right)\) | \(e\left(\frac{37}{274}\right)\) | \(e\left(\frac{529}{548}\right)\) | \(e\left(\frac{903}{1096}\right)\) | \(e\left(\frac{915}{1096}\right)\) |
| \(\chi_{1097}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{267}{274}\right)\) | \(e\left(\frac{855}{1096}\right)\) | \(e\left(\frac{130}{137}\right)\) | \(e\left(\frac{493}{1096}\right)\) | \(e\left(\frac{827}{1096}\right)\) | \(e\left(\frac{1053}{1096}\right)\) | \(e\left(\frac{253}{274}\right)\) | \(e\left(\frac{307}{548}\right)\) | \(e\left(\frac{465}{1096}\right)\) | \(e\left(\frac{1021}{1096}\right)\) |
| \(\chi_{1097}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{274}\right)\) | \(e\left(\frac{995}{1096}\right)\) | \(e\left(\frac{92}{137}\right)\) | \(e\left(\frac{625}{1096}\right)\) | \(e\left(\frac{815}{1096}\right)\) | \(e\left(\frac{937}{1096}\right)\) | \(e\left(\frac{139}{274}\right)\) | \(e\left(\frac{447}{548}\right)\) | \(e\left(\frac{445}{1096}\right)\) | \(e\left(\frac{105}{1096}\right)\) |
| \(\chi_{1097}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{274}\right)\) | \(e\left(\frac{135}{1096}\right)\) | \(e\left(\frac{71}{137}\right)\) | \(e\left(\frac{597}{1096}\right)\) | \(e\left(\frac{419}{1096}\right)\) | \(e\left(\frac{397}{1096}\right)\) | \(e\left(\frac{213}{274}\right)\) | \(e\left(\frac{135}{548}\right)\) | \(e\left(\frac{881}{1096}\right)\) | \(e\left(\frac{565}{1096}\right)\) |
| \(\chi_{1097}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{147}{274}\right)\) | \(e\left(\frac{951}{1096}\right)\) | \(e\left(\frac{10}{137}\right)\) | \(e\left(\frac{333}{1096}\right)\) | \(e\left(\frac{443}{1096}\right)\) | \(e\left(\frac{629}{1096}\right)\) | \(e\left(\frac{167}{274}\right)\) | \(e\left(\frac{403}{548}\right)\) | \(e\left(\frac{921}{1096}\right)\) | \(e\left(\frac{205}{1096}\right)\) |
| \(\chi_{1097}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{274}\right)\) | \(e\left(\frac{483}{1096}\right)\) | \(e\left(\frac{47}{137}\right)\) | \(e\left(\frac{17}{1096}\right)\) | \(e\left(\frac{671}{1096}\right)\) | \(e\left(\frac{641}{1096}\right)\) | \(e\left(\frac{141}{274}\right)\) | \(e\left(\frac{483}{548}\right)\) | \(e\left(\frac{205}{1096}\right)\) | \(e\left(\frac{73}{1096}\right)\) |
| \(\chi_{1097}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{274}\right)\) | \(e\left(\frac{557}{1096}\right)\) | \(e\left(\frac{23}{137}\right)\) | \(e\left(\frac{807}{1096}\right)\) | \(e\left(\frac{649}{1096}\right)\) | \(e\left(\frac{63}{1096}\right)\) | \(e\left(\frac{69}{274}\right)\) | \(e\left(\frac{9}{548}\right)\) | \(e\left(\frac{899}{1096}\right)\) | \(e\left(\frac{951}{1096}\right)\) |
| \(\chi_{1097}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{274}\right)\) | \(e\left(\frac{27}{1096}\right)\) | \(e\left(\frac{69}{137}\right)\) | \(e\left(\frac{777}{1096}\right)\) | \(e\left(\frac{303}{1096}\right)\) | \(e\left(\frac{737}{1096}\right)\) | \(e\left(\frac{207}{274}\right)\) | \(e\left(\frac{27}{548}\right)\) | \(e\left(\frac{1053}{1096}\right)\) | \(e\left(\frac{113}{1096}\right)\) |