from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(1092))
M = H._module
chi = DirichletCharacter(H, M([754,63]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,8281))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8281\) | |
| Conductor: | \(8281\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1092\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{1092})$ |
| Fixed field: | Number field defined by a degree 1092 polynomial (not computed) |
First 31 of 288 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8281}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{1092}\right)\) | \(e\left(\frac{461}{546}\right)\) | \(e\left(\frac{11}{546}\right)\) | \(e\left(\frac{593}{1092}\right)\) | \(e\left(\frac{311}{364}\right)\) | \(e\left(\frac{11}{364}\right)\) | \(e\left(\frac{188}{273}\right)\) | \(e\left(\frac{151}{273}\right)\) | \(e\left(\frac{613}{1092}\right)\) | \(e\left(\frac{236}{273}\right)\) |
| \(\chi_{8281}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{545}{1092}\right)\) | \(e\left(\frac{107}{546}\right)\) | \(e\left(\frac{545}{546}\right)\) | \(e\left(\frac{95}{1092}\right)\) | \(e\left(\frac{253}{364}\right)\) | \(e\left(\frac{181}{364}\right)\) | \(e\left(\frac{107}{273}\right)\) | \(e\left(\frac{160}{273}\right)\) | \(e\left(\frac{391}{1092}\right)\) | \(e\left(\frac{53}{273}\right)\) |
| \(\chi_{8281}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{253}{1092}\right)\) | \(e\left(\frac{229}{546}\right)\) | \(e\left(\frac{253}{546}\right)\) | \(e\left(\frac{535}{1092}\right)\) | \(e\left(\frac{237}{364}\right)\) | \(e\left(\frac{253}{364}\right)\) | \(e\left(\frac{229}{273}\right)\) | \(e\left(\frac{197}{273}\right)\) | \(e\left(\frac{995}{1092}\right)\) | \(e\left(\frac{241}{273}\right)\) |
| \(\chi_{8281}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1007}{1092}\right)\) | \(e\left(\frac{359}{546}\right)\) | \(e\left(\frac{461}{546}\right)\) | \(e\left(\frac{977}{1092}\right)\) | \(e\left(\frac{211}{364}\right)\) | \(e\left(\frac{279}{364}\right)\) | \(e\left(\frac{86}{273}\right)\) | \(e\left(\frac{223}{273}\right)\) | \(e\left(\frac{1021}{1092}\right)\) | \(e\left(\frac{137}{273}\right)\) |
| \(\chi_{8281}(122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{883}{1092}\right)\) | \(e\left(\frac{523}{546}\right)\) | \(e\left(\frac{337}{546}\right)\) | \(e\left(\frac{745}{1092}\right)\) | \(e\left(\frac{279}{364}\right)\) | \(e\left(\frac{155}{364}\right)\) | \(e\left(\frac{250}{273}\right)\) | \(e\left(\frac{134}{273}\right)\) | \(e\left(\frac{365}{1092}\right)\) | \(e\left(\frac{157}{273}\right)\) |
| \(\chi_{8281}(138,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{953}{1092}\right)\) | \(e\left(\frac{131}{546}\right)\) | \(e\left(\frac{407}{546}\right)\) | \(e\left(\frac{647}{1092}\right)\) | \(e\left(\frac{41}{364}\right)\) | \(e\left(\frac{225}{364}\right)\) | \(e\left(\frac{131}{273}\right)\) | \(e\left(\frac{127}{273}\right)\) | \(e\left(\frac{295}{1092}\right)\) | \(e\left(\frac{269}{273}\right)\) |
| \(\chi_{8281}(164,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{1092}\right)\) | \(e\left(\frac{409}{546}\right)\) | \(e\left(\frac{37}{546}\right)\) | \(e\left(\frac{307}{1092}\right)\) | \(e\left(\frac{285}{364}\right)\) | \(e\left(\frac{37}{364}\right)\) | \(e\left(\frac{136}{273}\right)\) | \(e\left(\frac{86}{273}\right)\) | \(e\left(\frac{275}{1092}\right)\) | \(e\left(\frac{223}{273}\right)\) |
| \(\chi_{8281}(187,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{911}{1092}\right)\) | \(e\left(\frac{257}{546}\right)\) | \(e\left(\frac{365}{546}\right)\) | \(e\left(\frac{269}{1092}\right)\) | \(e\left(\frac{111}{364}\right)\) | \(e\left(\frac{183}{364}\right)\) | \(e\left(\frac{257}{273}\right)\) | \(e\left(\frac{22}{273}\right)\) | \(e\left(\frac{337}{1092}\right)\) | \(e\left(\frac{38}{273}\right)\) |
| \(\chi_{8281}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{1092}\right)\) | \(e\left(\frac{31}{546}\right)\) | \(e\left(\frac{163}{546}\right)\) | \(e\left(\frac{349}{1092}\right)\) | \(e\left(\frac{75}{364}\right)\) | \(e\left(\frac{163}{364}\right)\) | \(e\left(\frac{31}{273}\right)\) | \(e\left(\frac{128}{273}\right)\) | \(e\left(\frac{149}{1092}\right)\) | \(e\left(\frac{97}{273}\right)\) |
| \(\chi_{8281}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{1092}\right)\) | \(e\left(\frac{155}{546}\right)\) | \(e\left(\frac{269}{546}\right)\) | \(e\left(\frac{107}{1092}\right)\) | \(e\left(\frac{193}{364}\right)\) | \(e\left(\frac{269}{364}\right)\) | \(e\left(\frac{155}{273}\right)\) | \(e\left(\frac{94}{273}\right)\) | \(e\left(\frac{199}{1092}\right)\) | \(e\left(\frac{212}{273}\right)\) |
| \(\chi_{8281}(255,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{913}{1092}\right)\) | \(e\left(\frac{43}{546}\right)\) | \(e\left(\frac{367}{546}\right)\) | \(e\left(\frac{79}{1092}\right)\) | \(e\left(\frac{333}{364}\right)\) | \(e\left(\frac{185}{364}\right)\) | \(e\left(\frac{43}{273}\right)\) | \(e\left(\frac{248}{273}\right)\) | \(e\left(\frac{647}{1092}\right)\) | \(e\left(\frac{205}{273}\right)\) |
| \(\chi_{8281}(278,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{815}{1092}\right)\) | \(e\left(\frac{155}{546}\right)\) | \(e\left(\frac{269}{546}\right)\) | \(e\left(\frac{653}{1092}\right)\) | \(e\left(\frac{11}{364}\right)\) | \(e\left(\frac{87}{364}\right)\) | \(e\left(\frac{155}{273}\right)\) | \(e\left(\frac{94}{273}\right)\) | \(e\left(\frac{745}{1092}\right)\) | \(e\left(\frac{212}{273}\right)\) |
| \(\chi_{8281}(304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{535}{1092}\right)\) | \(e\left(\frac{85}{546}\right)\) | \(e\left(\frac{535}{546}\right)\) | \(e\left(\frac{1045}{1092}\right)\) | \(e\left(\frac{235}{364}\right)\) | \(e\left(\frac{171}{364}\right)\) | \(e\left(\frac{85}{273}\right)\) | \(e\left(\frac{122}{273}\right)\) | \(e\left(\frac{1025}{1092}\right)\) | \(e\left(\frac{37}{273}\right)\) |
| \(\chi_{8281}(320,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{677}{1092}\right)\) | \(e\left(\frac{179}{546}\right)\) | \(e\left(\frac{131}{546}\right)\) | \(e\left(\frac{659}{1092}\right)\) | \(e\left(\frac{345}{364}\right)\) | \(e\left(\frac{313}{364}\right)\) | \(e\left(\frac{179}{273}\right)\) | \(e\left(\frac{61}{273}\right)\) | \(e\left(\frac{103}{1092}\right)\) | \(e\left(\frac{155}{273}\right)\) |
| \(\chi_{8281}(346,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{697}{1092}\right)\) | \(e\left(\frac{223}{546}\right)\) | \(e\left(\frac{151}{546}\right)\) | \(e\left(\frac{943}{1092}\right)\) | \(e\left(\frac{17}{364}\right)\) | \(e\left(\frac{333}{364}\right)\) | \(e\left(\frac{223}{273}\right)\) | \(e\left(\frac{137}{273}\right)\) | \(e\left(\frac{1019}{1092}\right)\) | \(e\left(\frac{187}{273}\right)\) |
| \(\chi_{8281}(369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{719}{1092}\right)\) | \(e\left(\frac{53}{546}\right)\) | \(e\left(\frac{173}{546}\right)\) | \(e\left(\frac{1037}{1092}\right)\) | \(e\left(\frac{275}{364}\right)\) | \(e\left(\frac{355}{364}\right)\) | \(e\left(\frac{53}{273}\right)\) | \(e\left(\frac{166}{273}\right)\) | \(e\left(\frac{61}{1092}\right)\) | \(e\left(\frac{113}{273}\right)\) |
| \(\chi_{8281}(395,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{907}{1092}\right)\) | \(e\left(\frac{139}{546}\right)\) | \(e\left(\frac{361}{546}\right)\) | \(e\left(\frac{649}{1092}\right)\) | \(e\left(\frac{31}{364}\right)\) | \(e\left(\frac{179}{364}\right)\) | \(e\left(\frac{139}{273}\right)\) | \(e\left(\frac{116}{273}\right)\) | \(e\left(\frac{809}{1092}\right)\) | \(e\left(\frac{250}{273}\right)\) |
| \(\chi_{8281}(486,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{1092}\right)\) | \(e\left(\frac{193}{546}\right)\) | \(e\left(\frac{187}{546}\right)\) | \(e\left(\frac{253}{1092}\right)\) | \(e\left(\frac{191}{364}\right)\) | \(e\left(\frac{187}{364}\right)\) | \(e\left(\frac{193}{273}\right)\) | \(e\left(\frac{110}{273}\right)\) | \(e\left(\frac{593}{1092}\right)\) | \(e\left(\frac{190}{273}\right)\) |
| \(\chi_{8281}(502,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{401}{1092}\right)\) | \(e\left(\frac{227}{546}\right)\) | \(e\left(\frac{401}{546}\right)\) | \(e\left(\frac{671}{1092}\right)\) | \(e\left(\frac{285}{364}\right)\) | \(e\left(\frac{37}{364}\right)\) | \(e\left(\frac{227}{273}\right)\) | \(e\left(\frac{268}{273}\right)\) | \(e\left(\frac{1003}{1092}\right)\) | \(e\left(\frac{41}{273}\right)\) |
| \(\chi_{8281}(528,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{1092}\right)\) | \(e\left(\frac{37}{546}\right)\) | \(e\left(\frac{265}{546}\right)\) | \(e\left(\frac{487}{1092}\right)\) | \(e\left(\frac{113}{364}\right)\) | \(e\left(\frac{265}{364}\right)\) | \(e\left(\frac{37}{273}\right)\) | \(e\left(\frac{188}{273}\right)\) | \(e\left(\frac{671}{1092}\right)\) | \(e\left(\frac{151}{273}\right)\) |
| \(\chi_{8281}(551,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{527}{1092}\right)\) | \(e\left(\frac{395}{546}\right)\) | \(e\left(\frac{527}{546}\right)\) | \(e\left(\frac{713}{1092}\right)\) | \(e\left(\frac{75}{364}\right)\) | \(e\left(\frac{163}{364}\right)\) | \(e\left(\frac{122}{273}\right)\) | \(e\left(\frac{37}{273}\right)\) | \(e\left(\frac{877}{1092}\right)\) | \(e\left(\frac{188}{273}\right)\) |
| \(\chi_{8281}(593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{809}{1092}\right)\) | \(e\left(\frac{251}{546}\right)\) | \(e\left(\frac{263}{546}\right)\) | \(e\left(\frac{131}{1092}\right)\) | \(e\left(\frac{73}{364}\right)\) | \(e\left(\frac{81}{364}\right)\) | \(e\left(\frac{251}{273}\right)\) | \(e\left(\frac{235}{273}\right)\) | \(e\left(\frac{907}{1092}\right)\) | \(e\left(\frac{257}{273}\right)\) |
| \(\chi_{8281}(642,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{431}{1092}\right)\) | \(e\left(\frac{293}{546}\right)\) | \(e\left(\frac{431}{546}\right)\) | \(e\left(\frac{5}{1092}\right)\) | \(e\left(\frac{339}{364}\right)\) | \(e\left(\frac{67}{364}\right)\) | \(e\left(\frac{20}{273}\right)\) | \(e\left(\frac{109}{273}\right)\) | \(e\left(\frac{193}{1092}\right)\) | \(e\left(\frac{89}{273}\right)\) |
| \(\chi_{8281}(684,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{1092}\right)\) | \(e\left(\frac{275}{546}\right)\) | \(e\left(\frac{125}{546}\right)\) | \(e\left(\frac{683}{1092}\right)\) | \(e\left(\frac{225}{364}\right)\) | \(e\left(\frac{125}{364}\right)\) | \(e\left(\frac{2}{273}\right)\) | \(e\left(\frac{202}{273}\right)\) | \(e\left(\frac{811}{1092}\right)\) | \(e\left(\frac{200}{273}\right)\) |
| \(\chi_{8281}(710,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{925}{1092}\right)\) | \(e\left(\frac{397}{546}\right)\) | \(e\left(\frac{379}{546}\right)\) | \(e\left(\frac{31}{1092}\right)\) | \(e\left(\frac{209}{364}\right)\) | \(e\left(\frac{197}{364}\right)\) | \(e\left(\frac{124}{273}\right)\) | \(e\left(\frac{239}{273}\right)\) | \(e\left(\frac{323}{1092}\right)\) | \(e\left(\frac{115}{273}\right)\) |
| \(\chi_{8281}(733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{335}{1092}\right)\) | \(e\left(\frac{191}{546}\right)\) | \(e\left(\frac{335}{546}\right)\) | \(e\left(\frac{389}{1092}\right)\) | \(e\left(\frac{239}{364}\right)\) | \(e\left(\frac{335}{364}\right)\) | \(e\left(\frac{191}{273}\right)\) | \(e\left(\frac{181}{273}\right)\) | \(e\left(\frac{601}{1092}\right)\) | \(e\left(\frac{263}{273}\right)\) |
| \(\chi_{8281}(759,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{1092}\right)\) | \(e\left(\frac{355}{546}\right)\) | \(e\left(\frac{211}{546}\right)\) | \(e\left(\frac{157}{1092}\right)\) | \(e\left(\frac{307}{364}\right)\) | \(e\left(\frac{211}{364}\right)\) | \(e\left(\frac{82}{273}\right)\) | \(e\left(\frac{92}{273}\right)\) | \(e\left(\frac{1037}{1092}\right)\) | \(e\left(\frac{10}{273}\right)\) |
| \(\chi_{8281}(801,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{709}{1092}\right)\) | \(e\left(\frac{31}{546}\right)\) | \(e\left(\frac{163}{546}\right)\) | \(e\left(\frac{895}{1092}\right)\) | \(e\left(\frac{257}{364}\right)\) | \(e\left(\frac{345}{364}\right)\) | \(e\left(\frac{31}{273}\right)\) | \(e\left(\frac{128}{273}\right)\) | \(e\left(\frac{695}{1092}\right)\) | \(e\left(\frac{97}{273}\right)\) |
| \(\chi_{8281}(824,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{1092}\right)\) | \(e\left(\frac{89}{546}\right)\) | \(e\left(\frac{239}{546}\right)\) | \(e\left(\frac{773}{1092}\right)\) | \(e\left(\frac{139}{364}\right)\) | \(e\left(\frac{239}{364}\right)\) | \(e\left(\frac{89}{273}\right)\) | \(e\left(\frac{253}{273}\right)\) | \(e\left(\frac{1009}{1092}\right)\) | \(e\left(\frac{164}{273}\right)\) |
| \(\chi_{8281}(850,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{583}{1092}\right)\) | \(e\left(\frac{409}{546}\right)\) | \(e\left(\frac{37}{546}\right)\) | \(e\left(\frac{853}{1092}\right)\) | \(e\left(\frac{103}{364}\right)\) | \(e\left(\frac{219}{364}\right)\) | \(e\left(\frac{136}{273}\right)\) | \(e\left(\frac{86}{273}\right)\) | \(e\left(\frac{821}{1092}\right)\) | \(e\left(\frac{223}{273}\right)\) |
| \(\chi_{8281}(866,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{941}{1092}\right)\) | \(e\left(\frac{323}{546}\right)\) | \(e\left(\frac{395}{546}\right)\) | \(e\left(\frac{695}{1092}\right)\) | \(e\left(\frac{165}{364}\right)\) | \(e\left(\frac{213}{364}\right)\) | \(e\left(\frac{50}{273}\right)\) | \(e\left(\frac{136}{273}\right)\) | \(e\left(\frac{619}{1092}\right)\) | \(e\left(\frac{86}{273}\right)\) |