from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(1092))
M = H._module
chi = DirichletCharacter(H, M([676,7]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,8281))
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: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8281}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{364}\right)\) | \(e\left(\frac{113}{273}\right)\) | \(e\left(\frac{37}{182}\right)\) | \(e\left(\frac{11}{1092}\right)\) | \(e\left(\frac{563}{1092}\right)\) | \(e\left(\frac{111}{364}\right)\) | \(e\left(\frac{226}{273}\right)\) | \(e\left(\frac{61}{546}\right)\) | \(e\left(\frac{461}{1092}\right)\) | \(e\left(\frac{337}{546}\right)\) |
| \(\chi_{8281}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{364}\right)\) | \(e\left(\frac{19}{273}\right)\) | \(e\left(\frac{3}{182}\right)\) | \(e\left(\frac{55}{1092}\right)\) | \(e\left(\frac{631}{1092}\right)\) | \(e\left(\frac{191}{364}\right)\) | \(e\left(\frac{38}{273}\right)\) | \(e\left(\frac{305}{546}\right)\) | \(e\left(\frac{121}{1092}\right)\) | \(e\left(\frac{47}{546}\right)\) |
| \(\chi_{8281}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{283}{364}\right)\) | \(e\left(\frac{215}{273}\right)\) | \(e\left(\frac{101}{182}\right)\) | \(e\left(\frac{881}{1092}\right)\) | \(e\left(\frac{617}{1092}\right)\) | \(e\left(\frac{121}{364}\right)\) | \(e\left(\frac{157}{273}\right)\) | \(e\left(\frac{319}{546}\right)\) | \(e\left(\frac{191}{1092}\right)\) | \(e\left(\frac{187}{546}\right)\) |
| \(\chi_{8281}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{364}\right)\) | \(e\left(\frac{178}{273}\right)\) | \(e\left(\frac{167}{182}\right)\) | \(e\left(\frac{817}{1092}\right)\) | \(e\left(\frac{121}{1092}\right)\) | \(e\left(\frac{137}{364}\right)\) | \(e\left(\frac{83}{273}\right)\) | \(e\left(\frac{113}{546}\right)\) | \(e\left(\frac{487}{1092}\right)\) | \(e\left(\frac{311}{546}\right)\) |
| \(\chi_{8281}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{321}{364}\right)\) | \(e\left(\frac{122}{273}\right)\) | \(e\left(\frac{139}{182}\right)\) | \(e\left(\frac{971}{1092}\right)\) | \(e\left(\frac{359}{1092}\right)\) | \(e\left(\frac{235}{364}\right)\) | \(e\left(\frac{244}{273}\right)\) | \(e\left(\frac{421}{546}\right)\) | \(e\left(\frac{389}{1092}\right)\) | \(e\left(\frac{115}{546}\right)\) |
| \(\chi_{8281}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{364}\right)\) | \(e\left(\frac{139}{273}\right)\) | \(e\left(\frac{89}{182}\right)\) | \(e\left(\frac{115}{1092}\right)\) | \(e\left(\frac{823}{1092}\right)\) | \(e\left(\frac{267}{364}\right)\) | \(e\left(\frac{5}{273}\right)\) | \(e\left(\frac{191}{546}\right)\) | \(e\left(\frac{253}{1092}\right)\) | \(e\left(\frac{545}{546}\right)\) |
| \(\chi_{8281}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{364}\right)\) | \(e\left(\frac{214}{273}\right)\) | \(e\left(\frac{29}{182}\right)\) | \(e\left(\frac{289}{1092}\right)\) | \(e\left(\frac{397}{1092}\right)\) | \(e\left(\frac{269}{364}\right)\) | \(e\left(\frac{155}{273}\right)\) | \(e\left(\frac{461}{546}\right)\) | \(e\left(\frac{199}{1092}\right)\) | \(e\left(\frac{515}{546}\right)\) |
| \(\chi_{8281}(184,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{364}\right)\) | \(e\left(\frac{131}{273}\right)\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{839}{1092}\right)\) | \(e\left(\frac{155}{1092}\right)\) | \(e\left(\frac{359}{364}\right)\) | \(e\left(\frac{262}{273}\right)\) | \(e\left(\frac{235}{546}\right)\) | \(e\left(\frac{317}{1092}\right)\) | \(e\left(\frac{439}{546}\right)\) |
| \(\chi_{8281}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{364}\right)\) | \(e\left(\frac{2}{273}\right)\) | \(e\left(\frac{53}{182}\right)\) | \(e\left(\frac{365}{1092}\right)\) | \(e\left(\frac{713}{1092}\right)\) | \(e\left(\frac{341}{364}\right)\) | \(e\left(\frac{4}{273}\right)\) | \(e\left(\frac{535}{546}\right)\) | \(e\left(\frac{803}{1092}\right)\) | \(e\left(\frac{163}{546}\right)\) |
| \(\chi_{8281}(228,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{255}{364}\right)\) | \(e\left(\frac{250}{273}\right)\) | \(e\left(\frac{73}{182}\right)\) | \(e\left(\frac{853}{1092}\right)\) | \(e\left(\frac{673}{1092}\right)\) | \(e\left(\frac{37}{364}\right)\) | \(e\left(\frac{227}{273}\right)\) | \(e\left(\frac{263}{546}\right)\) | \(e\left(\frac{1003}{1092}\right)\) | \(e\left(\frac{173}{546}\right)\) |
| \(\chi_{8281}(305,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{261}{364}\right)\) | \(e\left(\frac{106}{273}\right)\) | \(e\left(\frac{79}{182}\right)\) | \(e\left(\frac{235}{1092}\right)\) | \(e\left(\frac{115}{1092}\right)\) | \(e\left(\frac{55}{364}\right)\) | \(e\left(\frac{212}{273}\right)\) | \(e\left(\frac{509}{546}\right)\) | \(e\left(\frac{517}{1092}\right)\) | \(e\left(\frac{449}{546}\right)\) |
| \(\chi_{8281}(310,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{364}\right)\) | \(e\left(\frac{32}{273}\right)\) | \(e\left(\frac{29}{182}\right)\) | \(e\left(\frac{653}{1092}\right)\) | \(e\left(\frac{761}{1092}\right)\) | \(e\left(\frac{269}{364}\right)\) | \(e\left(\frac{64}{273}\right)\) | \(e\left(\frac{97}{546}\right)\) | \(e\left(\frac{563}{1092}\right)\) | \(e\left(\frac{151}{546}\right)\) |
| \(\chi_{8281}(366,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{364}\right)\) | \(e\left(\frac{149}{273}\right)\) | \(e\left(\frac{81}{182}\right)\) | \(e\left(\frac{575}{1092}\right)\) | \(e\left(\frac{839}{1092}\right)\) | \(e\left(\frac{243}{364}\right)\) | \(e\left(\frac{25}{273}\right)\) | \(e\left(\frac{409}{546}\right)\) | \(e\left(\frac{173}{1092}\right)\) | \(e\left(\frac{541}{546}\right)\) |
| \(\chi_{8281}(396,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{165}{364}\right)\) | \(e\left(\frac{226}{273}\right)\) | \(e\left(\frac{165}{182}\right)\) | \(e\left(\frac{295}{1092}\right)\) | \(e\left(\frac{307}{1092}\right)\) | \(e\left(\frac{131}{364}\right)\) | \(e\left(\frac{179}{273}\right)\) | \(e\left(\frac{395}{546}\right)\) | \(e\left(\frac{649}{1092}\right)\) | \(e\left(\frac{401}{546}\right)\) |
| \(\chi_{8281}(401,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{364}\right)\) | \(e\left(\frac{62}{273}\right)\) | \(e\left(\frac{5}{182}\right)\) | \(e\left(\frac{941}{1092}\right)\) | \(e\left(\frac{809}{1092}\right)\) | \(e\left(\frac{197}{364}\right)\) | \(e\left(\frac{124}{273}\right)\) | \(e\left(\frac{205}{546}\right)\) | \(e\left(\frac{323}{1092}\right)\) | \(e\left(\frac{139}{546}\right)\) |
| \(\chi_{8281}(457,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{364}\right)\) | \(e\left(\frac{158}{273}\right)\) | \(e\left(\frac{1}{182}\right)\) | \(e\left(\frac{443}{1092}\right)\) | \(e\left(\frac{635}{1092}\right)\) | \(e\left(\frac{3}{364}\right)\) | \(e\left(\frac{43}{273}\right)\) | \(e\left(\frac{223}{546}\right)\) | \(e\left(\frac{101}{1092}\right)\) | \(e\left(\frac{319}{546}\right)\) |
| \(\chi_{8281}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{364}\right)\) | \(e\left(\frac{73}{273}\right)\) | \(e\left(\frac{69}{182}\right)\) | \(e\left(\frac{355}{1092}\right)\) | \(e\left(\frac{499}{1092}\right)\) | \(e\left(\frac{207}{364}\right)\) | \(e\left(\frac{146}{273}\right)\) | \(e\left(\frac{281}{546}\right)\) | \(e\left(\frac{781}{1092}\right)\) | \(e\left(\frac{353}{546}\right)\) |
| \(\chi_{8281}(492,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{364}\right)\) | \(e\left(\frac{92}{273}\right)\) | \(e\left(\frac{163}{182}\right)\) | \(e\left(\frac{137}{1092}\right)\) | \(e\left(\frac{857}{1092}\right)\) | \(e\left(\frac{125}{364}\right)\) | \(e\left(\frac{184}{273}\right)\) | \(e\left(\frac{313}{546}\right)\) | \(e\left(\frac{83}{1092}\right)\) | \(e\left(\frac{127}{546}\right)\) |
| \(\chi_{8281}(501,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{364}\right)\) | \(e\left(\frac{85}{273}\right)\) | \(e\left(\frac{23}{182}\right)\) | \(e\left(\frac{361}{1092}\right)\) | \(e\left(\frac{409}{1092}\right)\) | \(e\left(\frac{69}{364}\right)\) | \(e\left(\frac{170}{273}\right)\) | \(e\left(\frac{215}{546}\right)\) | \(e\left(\frac{139}{1092}\right)\) | \(e\left(\frac{239}{546}\right)\) |
| \(\chi_{8281}(548,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{285}{364}\right)\) | \(e\left(\frac{167}{273}\right)\) | \(e\left(\frac{103}{182}\right)\) | \(e\left(\frac{311}{1092}\right)\) | \(e\left(\frac{431}{1092}\right)\) | \(e\left(\frac{127}{364}\right)\) | \(e\left(\frac{61}{273}\right)\) | \(e\left(\frac{37}{546}\right)\) | \(e\left(\frac{29}{1092}\right)\) | \(e\left(\frac{97}{546}\right)\) |
| \(\chi_{8281}(578,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{337}{364}\right)\) | \(e\left(\frac{193}{273}\right)\) | \(e\left(\frac{155}{182}\right)\) | \(e\left(\frac{415}{1092}\right)\) | \(e\left(\frac{691}{1092}\right)\) | \(e\left(\frac{283}{364}\right)\) | \(e\left(\frac{113}{273}\right)\) | \(e\left(\frac{167}{546}\right)\) | \(e\left(\frac{913}{1092}\right)\) | \(e\left(\frac{305}{546}\right)\) |
| \(\chi_{8281}(583,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{364}\right)\) | \(e\left(\frac{122}{273}\right)\) | \(e\left(\frac{139}{182}\right)\) | \(e\left(\frac{425}{1092}\right)\) | \(e\left(\frac{905}{1092}\right)\) | \(e\left(\frac{53}{364}\right)\) | \(e\left(\frac{244}{273}\right)\) | \(e\left(\frac{421}{546}\right)\) | \(e\left(\frac{935}{1092}\right)\) | \(e\left(\frac{115}{546}\right)\) |
| \(\chi_{8281}(592,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{364}\right)\) | \(e\left(\frac{121}{273}\right)\) | \(e\left(\frac{67}{182}\right)\) | \(e\left(\frac{925}{1092}\right)\) | \(e\left(\frac{685}{1092}\right)\) | \(e\left(\frac{201}{364}\right)\) | \(e\left(\frac{242}{273}\right)\) | \(e\left(\frac{17}{546}\right)\) | \(e\left(\frac{943}{1092}\right)\) | \(e\left(\frac{443}{546}\right)\) |
| \(\chi_{8281}(639,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{364}\right)\) | \(e\left(\frac{176}{273}\right)\) | \(e\left(\frac{23}{182}\right)\) | \(e\left(\frac{179}{1092}\right)\) | \(e\left(\frac{227}{1092}\right)\) | \(e\left(\frac{251}{364}\right)\) | \(e\left(\frac{79}{273}\right)\) | \(e\left(\frac{397}{546}\right)\) | \(e\left(\frac{1049}{1092}\right)\) | \(e\left(\frac{421}{546}\right)\) |
| \(\chi_{8281}(669,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{364}\right)\) | \(e\left(\frac{40}{273}\right)\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{475}{1092}\right)\) | \(e\left(\frac{883}{1092}\right)\) | \(e\left(\frac{359}{364}\right)\) | \(e\left(\frac{80}{273}\right)\) | \(e\left(\frac{53}{546}\right)\) | \(e\left(\frac{1045}{1092}\right)\) | \(e\left(\frac{257}{546}\right)\) |
| \(\chi_{8281}(674,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{364}\right)\) | \(e\left(\frac{152}{273}\right)\) | \(e\left(\frac{115}{182}\right)\) | \(e\left(\frac{713}{1092}\right)\) | \(e\left(\frac{953}{1092}\right)\) | \(e\left(\frac{345}{364}\right)\) | \(e\left(\frac{31}{273}\right)\) | \(e\left(\frac{529}{546}\right)\) | \(e\left(\frac{695}{1092}\right)\) | \(e\left(\frac{103}{546}\right)\) |
| \(\chi_{8281}(683,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{364}\right)\) | \(e\left(\frac{157}{273}\right)\) | \(e\left(\frac{111}{182}\right)\) | \(e\left(\frac{397}{1092}\right)\) | \(e\left(\frac{961}{1092}\right)\) | \(e\left(\frac{333}{364}\right)\) | \(e\left(\frac{41}{273}\right)\) | \(e\left(\frac{365}{546}\right)\) | \(e\left(\frac{655}{1092}\right)\) | \(e\left(\frac{101}{546}\right)\) |
| \(\chi_{8281}(730,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{364}\right)\) | \(e\left(\frac{185}{273}\right)\) | \(e\left(\frac{125}{182}\right)\) | \(e\left(\frac{47}{1092}\right)\) | \(e\left(\frac{23}{1092}\right)\) | \(e\left(\frac{11}{364}\right)\) | \(e\left(\frac{97}{273}\right)\) | \(e\left(\frac{211}{546}\right)\) | \(e\left(\frac{977}{1092}\right)\) | \(e\left(\frac{199}{546}\right)\) |
| \(\chi_{8281}(760,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{364}\right)\) | \(e\left(\frac{160}{273}\right)\) | \(e\left(\frac{145}{182}\right)\) | \(e\left(\frac{535}{1092}\right)\) | \(e\left(\frac{1075}{1092}\right)\) | \(e\left(\frac{71}{364}\right)\) | \(e\left(\frac{47}{273}\right)\) | \(e\left(\frac{485}{546}\right)\) | \(e\left(\frac{85}{1092}\right)\) | \(e\left(\frac{209}{546}\right)\) |
| \(\chi_{8281}(774,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{364}\right)\) | \(e\left(\frac{193}{273}\right)\) | \(e\left(\frac{155}{182}\right)\) | \(e\left(\frac{961}{1092}\right)\) | \(e\left(\frac{145}{1092}\right)\) | \(e\left(\frac{101}{364}\right)\) | \(e\left(\frac{113}{273}\right)\) | \(e\left(\frac{167}{546}\right)\) | \(e\left(\frac{367}{1092}\right)\) | \(e\left(\frac{305}{546}\right)\) |
| \(\chi_{8281}(821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{364}\right)\) | \(e\left(\frac{194}{273}\right)\) | \(e\left(\frac{45}{182}\right)\) | \(e\left(\frac{1007}{1092}\right)\) | \(e\left(\frac{911}{1092}\right)\) | \(e\left(\frac{135}{364}\right)\) | \(e\left(\frac{115}{273}\right)\) | \(e\left(\frac{25}{546}\right)\) | \(e\left(\frac{905}{1092}\right)\) | \(e\left(\frac{523}{546}\right)\) |