from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8027, base_ring=CyclotomicField(1276))
M = H._module
chi = DirichletCharacter(H, M([1044,99]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,8027))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8027\) | |
| Conductor: | \(8027\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1276\) | 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: | $\Q(\zeta_{1276})$ |
| Fixed field: | Number field defined by a degree 1276 polynomial (not computed) |
First 31 of 560 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8027}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{911}{1276}\right)\) | \(e\left(\frac{69}{638}\right)\) | \(e\left(\frac{273}{638}\right)\) | \(e\left(\frac{115}{638}\right)\) | \(e\left(\frac{1049}{1276}\right)\) | \(e\left(\frac{949}{1276}\right)\) | \(e\left(\frac{181}{1276}\right)\) | \(e\left(\frac{69}{319}\right)\) | \(e\left(\frac{1141}{1276}\right)\) | \(e\left(\frac{1267}{1276}\right)\) |
| \(\chi_{8027}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{707}{1276}\right)\) | \(e\left(\frac{375}{638}\right)\) | \(e\left(\frac{69}{638}\right)\) | \(e\left(\frac{625}{638}\right)\) | \(e\left(\frac{181}{1276}\right)\) | \(e\left(\frac{969}{1276}\right)\) | \(e\left(\frac{845}{1276}\right)\) | \(e\left(\frac{56}{319}\right)\) | \(e\left(\frac{681}{1276}\right)\) | \(e\left(\frac{811}{1276}\right)\) |
| \(\chi_{8027}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1165}{1276}\right)\) | \(e\left(\frac{7}{638}\right)\) | \(e\left(\frac{527}{638}\right)\) | \(e\left(\frac{437}{638}\right)\) | \(e\left(\frac{1179}{1276}\right)\) | \(e\left(\frac{799}{1276}\right)\) | \(e\left(\frac{943}{1276}\right)\) | \(e\left(\frac{7}{319}\right)\) | \(e\left(\frac{763}{1276}\right)\) | \(e\left(\frac{221}{1276}\right)\) |
| \(\chi_{8027}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{741}{1276}\right)\) | \(e\left(\frac{5}{638}\right)\) | \(e\left(\frac{103}{638}\right)\) | \(e\left(\frac{221}{638}\right)\) | \(e\left(\frac{751}{1276}\right)\) | \(e\left(\frac{115}{1276}\right)\) | \(e\left(\frac{947}{1276}\right)\) | \(e\left(\frac{5}{319}\right)\) | \(e\left(\frac{1183}{1276}\right)\) | \(e\left(\frac{249}{1276}\right)\) |
| \(\chi_{8027}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{537}{1276}\right)\) | \(e\left(\frac{311}{638}\right)\) | \(e\left(\frac{537}{638}\right)\) | \(e\left(\frac{93}{638}\right)\) | \(e\left(\frac{1159}{1276}\right)\) | \(e\left(\frac{135}{1276}\right)\) | \(e\left(\frac{335}{1276}\right)\) | \(e\left(\frac{311}{319}\right)\) | \(e\left(\frac{723}{1276}\right)\) | \(e\left(\frac{1069}{1276}\right)\) |
| \(\chi_{8027}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{307}{1276}\right)\) | \(e\left(\frac{337}{638}\right)\) | \(e\left(\frac{307}{638}\right)\) | \(e\left(\frac{349}{638}\right)\) | \(e\left(\frac{981}{1276}\right)\) | \(e\left(\frac{733}{1276}\right)\) | \(e\left(\frac{921}{1276}\right)\) | \(e\left(\frac{18}{319}\right)\) | \(e\left(\frac{1005}{1276}\right)\) | \(e\left(\frac{67}{1276}\right)\) |
| \(\chi_{8027}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{1276}\right)\) | \(e\left(\frac{113}{638}\right)\) | \(e\left(\frac{31}{638}\right)\) | \(e\left(\frac{401}{638}\right)\) | \(e\left(\frac{257}{1276}\right)\) | \(e\left(\frac{685}{1276}\right)\) | \(e\left(\frac{93}{1276}\right)\) | \(e\left(\frac{113}{319}\right)\) | \(e\left(\frac{833}{1276}\right)\) | \(e\left(\frac{651}{1276}\right)\) |
| \(\chi_{8027}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1105}{1276}\right)\) | \(e\left(\frac{97}{638}\right)\) | \(e\left(\frac{467}{638}\right)\) | \(e\left(\frac{587}{638}\right)\) | \(e\left(\frac{23}{1276}\right)\) | \(e\left(\frac{955}{1276}\right)\) | \(e\left(\frac{763}{1276}\right)\) | \(e\left(\frac{97}{319}\right)\) | \(e\left(\frac{1003}{1276}\right)\) | \(e\left(\frac{237}{1276}\right)\) |
| \(\chi_{8027}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{967}{1276}\right)\) | \(e\left(\frac{623}{638}\right)\) | \(e\left(\frac{329}{638}\right)\) | \(e\left(\frac{613}{638}\right)\) | \(e\left(\frac{937}{1276}\right)\) | \(e\left(\frac{293}{1276}\right)\) | \(e\left(\frac{349}{1276}\right)\) | \(e\left(\frac{304}{319}\right)\) | \(e\left(\frac{917}{1276}\right)\) | \(e\left(\frac{1167}{1276}\right)\) |
| \(\chi_{8027}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{917}{1276}\right)\) | \(e\left(\frac{379}{638}\right)\) | \(e\left(\frac{279}{638}\right)\) | \(e\left(\frac{419}{638}\right)\) | \(e\left(\frac{399}{1276}\right)\) | \(e\left(\frac{423}{1276}\right)\) | \(e\left(\frac{199}{1276}\right)\) | \(e\left(\frac{60}{319}\right)\) | \(e\left(\frac{479}{1276}\right)\) | \(e\left(\frac{117}{1276}\right)\) |
| \(\chi_{8027}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1051}{1276}\right)\) | \(e\left(\frac{497}{638}\right)\) | \(e\left(\frac{413}{638}\right)\) | \(e\left(\frac{403}{638}\right)\) | \(e\left(\frac{769}{1276}\right)\) | \(e\left(\frac{585}{1276}\right)\) | \(e\left(\frac{601}{1276}\right)\) | \(e\left(\frac{178}{319}\right)\) | \(e\left(\frac{581}{1276}\right)\) | \(e\left(\frac{379}{1276}\right)\) |
| \(\chi_{8027}(146,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{839}{1276}\right)\) | \(e\left(\frac{177}{638}\right)\) | \(e\left(\frac{201}{638}\right)\) | \(e\left(\frac{295}{638}\right)\) | \(e\left(\frac{1193}{1276}\right)\) | \(e\left(\frac{881}{1276}\right)\) | \(e\left(\frac{1241}{1276}\right)\) | \(e\left(\frac{177}{319}\right)\) | \(e\left(\frac{153}{1276}\right)\) | \(e\left(\frac{1031}{1276}\right)\) |
| \(\chi_{8027}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{1276}\right)\) | \(e\left(\frac{81}{638}\right)\) | \(e\left(\frac{265}{638}\right)\) | \(e\left(\frac{135}{638}\right)\) | \(e\left(\frac{427}{1276}\right)\) | \(e\left(\frac{587}{1276}\right)\) | \(e\left(\frac{795}{1276}\right)\) | \(e\left(\frac{81}{319}\right)\) | \(e\left(\frac{535}{1276}\right)\) | \(e\left(\frac{461}{1276}\right)\) |
| \(\chi_{8027}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{1276}\right)\) | \(e\left(\frac{267}{638}\right)\) | \(e\left(\frac{141}{638}\right)\) | \(e\left(\frac{445}{638}\right)\) | \(e\left(\frac{675}{1276}\right)\) | \(e\left(\frac{399}{1276}\right)\) | \(e\left(\frac{423}{1276}\right)\) | \(e\left(\frac{267}{319}\right)\) | \(e\left(\frac{1031}{1276}\right)\) | \(e\left(\frac{409}{1276}\right)\) |
| \(\chi_{8027}(170,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{709}{1276}\right)\) | \(e\left(\frac{53}{638}\right)\) | \(e\left(\frac{71}{638}\right)\) | \(e\left(\frac{301}{638}\right)\) | \(e\left(\frac{815}{1276}\right)\) | \(e\left(\frac{1219}{1276}\right)\) | \(e\left(\frac{851}{1276}\right)\) | \(e\left(\frac{53}{319}\right)\) | \(e\left(\frac{35}{1276}\right)\) | \(e\left(\frac{853}{1276}\right)\) |
| \(\chi_{8027}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{303}{1276}\right)\) | \(e\left(\frac{343}{638}\right)\) | \(e\left(\frac{303}{638}\right)\) | \(e\left(\frac{359}{638}\right)\) | \(e\left(\frac{989}{1276}\right)\) | \(e\left(\frac{233}{1276}\right)\) | \(e\left(\frac{909}{1276}\right)\) | \(e\left(\frac{24}{319}\right)\) | \(e\left(\frac{1021}{1276}\right)\) | \(e\left(\frac{1259}{1276}\right)\) |
| \(\chi_{8027}(186,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{903}{1276}\right)\) | \(e\left(\frac{81}{638}\right)\) | \(e\left(\frac{265}{638}\right)\) | \(e\left(\frac{135}{638}\right)\) | \(e\left(\frac{1065}{1276}\right)\) | \(e\left(\frac{1225}{1276}\right)\) | \(e\left(\frac{157}{1276}\right)\) | \(e\left(\frac{81}{319}\right)\) | \(e\left(\frac{1173}{1276}\right)\) | \(e\left(\frac{1099}{1276}\right)\) |
| \(\chi_{8027}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{327}{1276}\right)\) | \(e\left(\frac{307}{638}\right)\) | \(e\left(\frac{327}{638}\right)\) | \(e\left(\frac{299}{638}\right)\) | \(e\left(\frac{941}{1276}\right)\) | \(e\left(\frac{681}{1276}\right)\) | \(e\left(\frac{981}{1276}\right)\) | \(e\left(\frac{307}{319}\right)\) | \(e\left(\frac{925}{1276}\right)\) | \(e\left(\frac{487}{1276}\right)\) |
| \(\chi_{8027}(216,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{1276}\right)\) | \(e\left(\frac{207}{638}\right)\) | \(e\left(\frac{181}{638}\right)\) | \(e\left(\frac{345}{638}\right)\) | \(e\left(\frac{595}{1276}\right)\) | \(e\left(\frac{295}{1276}\right)\) | \(e\left(\frac{543}{1276}\right)\) | \(e\left(\frac{207}{319}\right)\) | \(e\left(\frac{871}{1276}\right)\) | \(e\left(\frac{1249}{1276}\right)\) |
| \(\chi_{8027}(246,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{719}{1276}\right)\) | \(e\left(\frac{357}{638}\right)\) | \(e\left(\frac{81}{638}\right)\) | \(e\left(\frac{595}{638}\right)\) | \(e\left(\frac{157}{1276}\right)\) | \(e\left(\frac{1193}{1276}\right)\) | \(e\left(\frac{881}{1276}\right)\) | \(e\left(\frac{38}{319}\right)\) | \(e\left(\frac{633}{1276}\right)\) | \(e\left(\frac{1063}{1276}\right)\) |
| \(\chi_{8027}(248,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{699}{1276}\right)\) | \(e\left(\frac{387}{638}\right)\) | \(e\left(\frac{61}{638}\right)\) | \(e\left(\frac{7}{638}\right)\) | \(e\left(\frac{197}{1276}\right)\) | \(e\left(\frac{1245}{1276}\right)\) | \(e\left(\frac{821}{1276}\right)\) | \(e\left(\frac{68}{319}\right)\) | \(e\left(\frac{713}{1276}\right)\) | \(e\left(\frac{643}{1276}\right)\) |
| \(\chi_{8027}(284,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{927}{1276}\right)\) | \(e\left(\frac{45}{638}\right)\) | \(e\left(\frac{289}{638}\right)\) | \(e\left(\frac{75}{638}\right)\) | \(e\left(\frac{1017}{1276}\right)\) | \(e\left(\frac{397}{1276}\right)\) | \(e\left(\frac{229}{1276}\right)\) | \(e\left(\frac{45}{319}\right)\) | \(e\left(\frac{1077}{1276}\right)\) | \(e\left(\frac{327}{1276}\right)\) |
| \(\chi_{8027}(288,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1253}{1276}\right)\) | \(e\left(\frac{513}{638}\right)\) | \(e\left(\frac{615}{638}\right)\) | \(e\left(\frac{217}{638}\right)\) | \(e\left(\frac{1003}{1276}\right)\) | \(e\left(\frac{315}{1276}\right)\) | \(e\left(\frac{1207}{1276}\right)\) | \(e\left(\frac{194}{319}\right)\) | \(e\left(\frac{411}{1276}\right)\) | \(e\left(\frac{793}{1276}\right)\) |
| \(\chi_{8027}(302,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{1276}\right)\) | \(e\left(\frac{153}{638}\right)\) | \(e\left(\frac{217}{638}\right)\) | \(e\left(\frac{255}{638}\right)\) | \(e\left(\frac{523}{1276}\right)\) | \(e\left(\frac{967}{1276}\right)\) | \(e\left(\frac{651}{1276}\right)\) | \(e\left(\frac{153}{319}\right)\) | \(e\left(\frac{727}{1276}\right)\) | \(e\left(\frac{729}{1276}\right)\) |
| \(\chi_{8027}(311,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1099}{1276}\right)\) | \(e\left(\frac{425}{638}\right)\) | \(e\left(\frac{461}{638}\right)\) | \(e\left(\frac{283}{638}\right)\) | \(e\left(\frac{673}{1276}\right)\) | \(e\left(\frac{205}{1276}\right)\) | \(e\left(\frac{745}{1276}\right)\) | \(e\left(\frac{106}{319}\right)\) | \(e\left(\frac{389}{1276}\right)\) | \(e\left(\frac{111}{1276}\right)\) |
| \(\chi_{8027}(328,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{515}{1276}\right)\) | \(e\left(\frac{25}{638}\right)\) | \(e\left(\frac{515}{638}\right)\) | \(e\left(\frac{467}{638}\right)\) | \(e\left(\frac{565}{1276}\right)\) | \(e\left(\frac{1213}{1276}\right)\) | \(e\left(\frac{269}{1276}\right)\) | \(e\left(\frac{25}{319}\right)\) | \(e\left(\frac{173}{1276}\right)\) | \(e\left(\frac{607}{1276}\right)\) |
| \(\chi_{8027}(338,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{1276}\right)\) | \(e\left(\frac{247}{638}\right)\) | \(e\left(\frac{367}{638}\right)\) | \(e\left(\frac{199}{638}\right)\) | \(e\left(\frac{861}{1276}\right)\) | \(e\left(\frac{577}{1276}\right)\) | \(e\left(\frac{1101}{1276}\right)\) | \(e\left(\frac{247}{319}\right)\) | \(e\left(\frac{765}{1276}\right)\) | \(e\left(\frac{51}{1276}\right)\) |
| \(\chi_{8027}(357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1055}{1276}\right)\) | \(e\left(\frac{491}{638}\right)\) | \(e\left(\frac{417}{638}\right)\) | \(e\left(\frac{393}{638}\right)\) | \(e\left(\frac{761}{1276}\right)\) | \(e\left(\frac{1085}{1276}\right)\) | \(e\left(\frac{613}{1276}\right)\) | \(e\left(\frac{172}{319}\right)\) | \(e\left(\frac{565}{1276}\right)\) | \(e\left(\frac{463}{1276}\right)\) |
| \(\chi_{8027}(370,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{573}{1276}\right)\) | \(e\left(\frac{257}{638}\right)\) | \(e\left(\frac{573}{638}\right)\) | \(e\left(\frac{3}{638}\right)\) | \(e\left(\frac{1087}{1276}\right)\) | \(e\left(\frac{807}{1276}\right)\) | \(e\left(\frac{443}{1276}\right)\) | \(e\left(\frac{257}{319}\right)\) | \(e\left(\frac{579}{1276}\right)\) | \(e\left(\frac{549}{1276}\right)\) |
| \(\chi_{8027}(377,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{1276}\right)\) | \(e\left(\frac{273}{638}\right)\) | \(e\left(\frac{137}{638}\right)\) | \(e\left(\frac{455}{638}\right)\) | \(e\left(\frac{683}{1276}\right)\) | \(e\left(\frac{1175}{1276}\right)\) | \(e\left(\frac{411}{1276}\right)\) | \(e\left(\frac{273}{319}\right)\) | \(e\left(\frac{1047}{1276}\right)\) | \(e\left(\frac{325}{1276}\right)\) |
| \(\chi_{8027}(384,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1049}{1276}\right)\) | \(e\left(\frac{181}{638}\right)\) | \(e\left(\frac{411}{638}\right)\) | \(e\left(\frac{89}{638}\right)\) | \(e\left(\frac{135}{1276}\right)\) | \(e\left(\frac{335}{1276}\right)\) | \(e\left(\frac{595}{1276}\right)\) | \(e\left(\frac{181}{319}\right)\) | \(e\left(\frac{1227}{1276}\right)\) | \(e\left(\frac{337}{1276}\right)\) |