from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1277, base_ring=CyclotomicField(1276))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1277))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1277\) | |
| Conductor: | \(1277\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1277}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1276}\right)\) | \(e\left(\frac{669}{1276}\right)\) | \(e\left(\frac{1}{638}\right)\) | \(e\left(\frac{661}{1276}\right)\) | \(e\left(\frac{335}{638}\right)\) | \(e\left(\frac{1113}{1276}\right)\) | \(e\left(\frac{3}{1276}\right)\) | \(e\left(\frac{31}{638}\right)\) | \(e\left(\frac{331}{638}\right)\) | \(e\left(\frac{61}{319}\right)\) |
| \(\chi_{1277}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{669}{1276}\right)\) | \(e\left(\frac{961}{1276}\right)\) | \(e\left(\frac{31}{638}\right)\) | \(e\left(\frac{713}{1276}\right)\) | \(e\left(\frac{177}{638}\right)\) | \(e\left(\frac{689}{1276}\right)\) | \(e\left(\frac{731}{1276}\right)\) | \(e\left(\frac{323}{638}\right)\) | \(e\left(\frac{53}{638}\right)\) | \(e\left(\frac{296}{319}\right)\) |
| \(\chi_{1277}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{661}{1276}\right)\) | \(e\left(\frac{713}{1276}\right)\) | \(e\left(\frac{23}{638}\right)\) | \(e\left(\frac{529}{1276}\right)\) | \(e\left(\frac{49}{638}\right)\) | \(e\left(\frac{717}{1276}\right)\) | \(e\left(\frac{707}{1276}\right)\) | \(e\left(\frac{75}{638}\right)\) | \(e\left(\frac{595}{638}\right)\) | \(e\left(\frac{127}{319}\right)\) |
| \(\chi_{1277}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1113}{1276}\right)\) | \(e\left(\frac{689}{1276}\right)\) | \(e\left(\frac{475}{638}\right)\) | \(e\left(\frac{717}{1276}\right)\) | \(e\left(\frac{263}{638}\right)\) | \(e\left(\frac{1049}{1276}\right)\) | \(e\left(\frac{787}{1276}\right)\) | \(e\left(\frac{51}{638}\right)\) | \(e\left(\frac{277}{638}\right)\) | \(e\left(\frac{265}{319}\right)\) |
| \(\chi_{1277}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{1276}\right)\) | \(e\left(\frac{731}{1276}\right)\) | \(e\left(\frac{3}{638}\right)\) | \(e\left(\frac{707}{1276}\right)\) | \(e\left(\frac{367}{638}\right)\) | \(e\left(\frac{787}{1276}\right)\) | \(e\left(\frac{9}{1276}\right)\) | \(e\left(\frac{93}{638}\right)\) | \(e\left(\frac{355}{638}\right)\) | \(e\left(\frac{183}{319}\right)\) |
| \(\chi_{1277}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{1276}\right)\) | \(e\left(\frac{39}{1276}\right)\) | \(e\left(\frac{63}{638}\right)\) | \(e\left(\frac{811}{1276}\right)\) | \(e\left(\frac{51}{638}\right)\) | \(e\left(\frac{1215}{1276}\right)\) | \(e\left(\frac{189}{1276}\right)\) | \(e\left(\frac{39}{638}\right)\) | \(e\left(\frac{437}{638}\right)\) | \(e\left(\frac{15}{319}\right)\) |
| \(\chi_{1277}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{663}{1276}\right)\) | \(e\left(\frac{775}{1276}\right)\) | \(e\left(\frac{25}{638}\right)\) | \(e\left(\frac{575}{1276}\right)\) | \(e\left(\frac{81}{638}\right)\) | \(e\left(\frac{391}{1276}\right)\) | \(e\left(\frac{713}{1276}\right)\) | \(e\left(\frac{137}{638}\right)\) | \(e\left(\frac{619}{638}\right)\) | \(e\left(\frac{249}{319}\right)\) |
| \(\chi_{1277}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{245}{1276}\right)\) | \(e\left(\frac{577}{1276}\right)\) | \(e\left(\frac{245}{638}\right)\) | \(e\left(\frac{1169}{1276}\right)\) | \(e\left(\frac{411}{638}\right)\) | \(e\left(\frac{897}{1276}\right)\) | \(e\left(\frac{735}{1276}\right)\) | \(e\left(\frac{577}{638}\right)\) | \(e\left(\frac{69}{638}\right)\) | \(e\left(\frac{271}{319}\right)\) |
| \(\chi_{1277}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{357}{1276}\right)\) | \(e\left(\frac{221}{1276}\right)\) | \(e\left(\frac{357}{638}\right)\) | \(e\left(\frac{1193}{1276}\right)\) | \(e\left(\frac{289}{638}\right)\) | \(e\left(\frac{505}{1276}\right)\) | \(e\left(\frac{1071}{1276}\right)\) | \(e\left(\frac{221}{638}\right)\) | \(e\left(\frac{137}{638}\right)\) | \(e\left(\frac{85}{319}\right)\) |
| \(\chi_{1277}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{731}{1276}\right)\) | \(e\left(\frac{331}{1276}\right)\) | \(e\left(\frac{93}{638}\right)\) | \(e\left(\frac{863}{1276}\right)\) | \(e\left(\frac{531}{638}\right)\) | \(e\left(\frac{791}{1276}\right)\) | \(e\left(\frac{917}{1276}\right)\) | \(e\left(\frac{331}{638}\right)\) | \(e\left(\frac{159}{638}\right)\) | \(e\left(\frac{250}{319}\right)\) |
| \(\chi_{1277}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1115}{1276}\right)\) | \(e\left(\frac{751}{1276}\right)\) | \(e\left(\frac{477}{638}\right)\) | \(e\left(\frac{763}{1276}\right)\) | \(e\left(\frac{295}{638}\right)\) | \(e\left(\frac{723}{1276}\right)\) | \(e\left(\frac{793}{1276}\right)\) | \(e\left(\frac{113}{638}\right)\) | \(e\left(\frac{301}{638}\right)\) | \(e\left(\frac{68}{319}\right)\) |
| \(\chi_{1277}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{815}{1276}\right)\) | \(e\left(\frac{383}{1276}\right)\) | \(e\left(\frac{177}{638}\right)\) | \(e\left(\frac{243}{1276}\right)\) | \(e\left(\frac{599}{638}\right)\) | \(e\left(\frac{1135}{1276}\right)\) | \(e\left(\frac{1169}{1276}\right)\) | \(e\left(\frac{383}{638}\right)\) | \(e\left(\frac{529}{638}\right)\) | \(e\left(\frac{270}{319}\right)\) |
| \(\chi_{1277}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{1276}\right)\) | \(e\left(\frac{793}{1276}\right)\) | \(e\left(\frac{5}{638}\right)\) | \(e\left(\frac{753}{1276}\right)\) | \(e\left(\frac{399}{638}\right)\) | \(e\left(\frac{461}{1276}\right)\) | \(e\left(\frac{15}{1276}\right)\) | \(e\left(\frac{155}{638}\right)\) | \(e\left(\frac{379}{638}\right)\) | \(e\left(\frac{305}{319}\right)\) |
| \(\chi_{1277}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1201}{1276}\right)\) | \(e\left(\frac{865}{1276}\right)\) | \(e\left(\frac{563}{638}\right)\) | \(e\left(\frac{189}{1276}\right)\) | \(e\left(\frac{395}{638}\right)\) | \(e\left(\frac{741}{1276}\right)\) | \(e\left(\frac{1051}{1276}\right)\) | \(e\left(\frac{227}{638}\right)\) | \(e\left(\frac{57}{638}\right)\) | \(e\left(\frac{210}{319}\right)\) |
| \(\chi_{1277}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{1276}\right)\) | \(e\left(\frac{809}{1276}\right)\) | \(e\left(\frac{129}{638}\right)\) | \(e\left(\frac{1053}{1276}\right)\) | \(e\left(\frac{469}{638}\right)\) | \(e\left(\frac{665}{1276}\right)\) | \(e\left(\frac{387}{1276}\right)\) | \(e\left(\frac{171}{638}\right)\) | \(e\left(\frac{591}{638}\right)\) | \(e\left(\frac{213}{319}\right)\) |
| \(\chi_{1277}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1025}{1276}\right)\) | \(e\left(\frac{513}{1276}\right)\) | \(e\left(\frac{387}{638}\right)\) | \(e\left(\frac{1245}{1276}\right)\) | \(e\left(\frac{131}{638}\right)\) | \(e\left(\frac{81}{1276}\right)\) | \(e\left(\frac{523}{1276}\right)\) | \(e\left(\frac{513}{638}\right)\) | \(e\left(\frac{497}{638}\right)\) | \(e\left(\frac{1}{319}\right)\) |
| \(\chi_{1277}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{507}{1276}\right)\) | \(e\left(\frac{1043}{1276}\right)\) | \(e\left(\frac{507}{638}\right)\) | \(e\left(\frac{815}{1276}\right)\) | \(e\left(\frac{137}{638}\right)\) | \(e\left(\frac{299}{1276}\right)\) | \(e\left(\frac{245}{1276}\right)\) | \(e\left(\frac{405}{638}\right)\) | \(e\left(\frac{23}{638}\right)\) | \(e\left(\frac{303}{319}\right)\) |
| \(\chi_{1277}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{301}{1276}\right)\) | \(e\left(\frac{1037}{1276}\right)\) | \(e\left(\frac{301}{638}\right)\) | \(e\left(\frac{1181}{1276}\right)\) | \(e\left(\frac{31}{638}\right)\) | \(e\left(\frac{701}{1276}\right)\) | \(e\left(\frac{903}{1276}\right)\) | \(e\left(\frac{399}{638}\right)\) | \(e\left(\frac{103}{638}\right)\) | \(e\left(\frac{178}{319}\right)\) |
| \(\chi_{1277}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{723}{1276}\right)\) | \(e\left(\frac{83}{1276}\right)\) | \(e\left(\frac{85}{638}\right)\) | \(e\left(\frac{679}{1276}\right)\) | \(e\left(\frac{403}{638}\right)\) | \(e\left(\frac{819}{1276}\right)\) | \(e\left(\frac{893}{1276}\right)\) | \(e\left(\frac{83}{638}\right)\) | \(e\left(\frac{63}{638}\right)\) | \(e\left(\frac{81}{319}\right)\) |
| \(\chi_{1277}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{405}{1276}\right)\) | \(e\left(\frac{433}{1276}\right)\) | \(e\left(\frac{405}{638}\right)\) | \(e\left(\frac{1021}{1276}\right)\) | \(e\left(\frac{419}{638}\right)\) | \(e\left(\frac{337}{1276}\right)\) | \(e\left(\frac{1215}{1276}\right)\) | \(e\left(\frac{433}{638}\right)\) | \(e\left(\frac{75}{638}\right)\) | \(e\left(\frac{142}{319}\right)\) |
| \(\chi_{1277}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{673}{1276}\right)\) | \(e\left(\frac{1085}{1276}\right)\) | \(e\left(\frac{35}{638}\right)\) | \(e\left(\frac{805}{1276}\right)\) | \(e\left(\frac{241}{638}\right)\) | \(e\left(\frac{37}{1276}\right)\) | \(e\left(\frac{743}{1276}\right)\) | \(e\left(\frac{447}{638}\right)\) | \(e\left(\frac{101}{638}\right)\) | \(e\left(\frac{221}{319}\right)\) |
| \(\chi_{1277}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{1276}\right)\) | \(e\left(\frac{819}{1276}\right)\) | \(e\left(\frac{47}{638}\right)\) | \(e\left(\frac{443}{1276}\right)\) | \(e\left(\frac{433}{638}\right)\) | \(e\left(\frac{1271}{1276}\right)\) | \(e\left(\frac{141}{1276}\right)\) | \(e\left(\frac{181}{638}\right)\) | \(e\left(\frac{245}{638}\right)\) | \(e\left(\frac{315}{319}\right)\) |
| \(\chi_{1277}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{593}{1276}\right)\) | \(e\left(\frac{1157}{1276}\right)\) | \(e\left(\frac{593}{638}\right)\) | \(e\left(\frac{241}{1276}\right)\) | \(e\left(\frac{237}{638}\right)\) | \(e\left(\frac{317}{1276}\right)\) | \(e\left(\frac{503}{1276}\right)\) | \(e\left(\frac{519}{638}\right)\) | \(e\left(\frac{417}{638}\right)\) | \(e\left(\frac{126}{319}\right)\) |
| \(\chi_{1277}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{905}{1276}\right)\) | \(e\left(\frac{621}{1276}\right)\) | \(e\left(\frac{267}{638}\right)\) | \(e\left(\frac{1037}{1276}\right)\) | \(e\left(\frac{125}{638}\right)\) | \(e\left(\frac{501}{1276}\right)\) | \(e\left(\frac{163}{1276}\right)\) | \(e\left(\frac{621}{638}\right)\) | \(e\left(\frac{333}{638}\right)\) | \(e\left(\frac{18}{319}\right)\) |
| \(\chi_{1277}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{797}{1276}\right)\) | \(e\left(\frac{1101}{1276}\right)\) | \(e\left(\frac{159}{638}\right)\) | \(e\left(\frac{1105}{1276}\right)\) | \(e\left(\frac{311}{638}\right)\) | \(e\left(\frac{241}{1276}\right)\) | \(e\left(\frac{1115}{1276}\right)\) | \(e\left(\frac{463}{638}\right)\) | \(e\left(\frac{313}{638}\right)\) | \(e\left(\frac{129}{319}\right)\) |
| \(\chi_{1277}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{393}{1276}\right)\) | \(e\left(\frac{61}{1276}\right)\) | \(e\left(\frac{393}{638}\right)\) | \(e\left(\frac{745}{1276}\right)\) | \(e\left(\frac{227}{638}\right)\) | \(e\left(\frac{1017}{1276}\right)\) | \(e\left(\frac{1179}{1276}\right)\) | \(e\left(\frac{61}{638}\right)\) | \(e\left(\frac{569}{638}\right)\) | \(e\left(\frac{48}{319}\right)\) |
| \(\chi_{1277}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1175}{1276}\right)\) | \(e\left(\frac{59}{1276}\right)\) | \(e\left(\frac{537}{638}\right)\) | \(e\left(\frac{867}{1276}\right)\) | \(e\left(\frac{617}{638}\right)\) | \(e\left(\frac{1151}{1276}\right)\) | \(e\left(\frac{973}{1276}\right)\) | \(e\left(\frac{59}{638}\right)\) | \(e\left(\frac{383}{638}\right)\) | \(e\left(\frac{219}{319}\right)\) |
| \(\chi_{1277}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1017}{1276}\right)\) | \(e\left(\frac{265}{1276}\right)\) | \(e\left(\frac{379}{638}\right)\) | \(e\left(\frac{1061}{1276}\right)\) | \(e\left(\frac{3}{638}\right)\) | \(e\left(\frac{109}{1276}\right)\) | \(e\left(\frac{499}{1276}\right)\) | \(e\left(\frac{265}{638}\right)\) | \(e\left(\frac{401}{638}\right)\) | \(e\left(\frac{151}{319}\right)\) |
| \(\chi_{1277}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{499}{1276}\right)\) | \(e\left(\frac{795}{1276}\right)\) | \(e\left(\frac{499}{638}\right)\) | \(e\left(\frac{631}{1276}\right)\) | \(e\left(\frac{9}{638}\right)\) | \(e\left(\frac{327}{1276}\right)\) | \(e\left(\frac{221}{1276}\right)\) | \(e\left(\frac{157}{638}\right)\) | \(e\left(\frac{565}{638}\right)\) | \(e\left(\frac{134}{319}\right)\) |
| \(\chi_{1277}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1213}{1276}\right)\) | \(e\left(\frac{1237}{1276}\right)\) | \(e\left(\frac{575}{638}\right)\) | \(e\left(\frac{465}{1276}\right)\) | \(e\left(\frac{587}{638}\right)\) | \(e\left(\frac{61}{1276}\right)\) | \(e\left(\frac{1087}{1276}\right)\) | \(e\left(\frac{599}{638}\right)\) | \(e\left(\frac{201}{638}\right)\) | \(e\left(\frac{304}{319}\right)\) |
| \(\chi_{1277}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{1276}\right)\) | \(e\left(\frac{101}{1276}\right)\) | \(e\left(\frac{65}{638}\right)\) | \(e\left(\frac{857}{1276}\right)\) | \(e\left(\frac{83}{638}\right)\) | \(e\left(\frac{889}{1276}\right)\) | \(e\left(\frac{195}{1276}\right)\) | \(e\left(\frac{101}{638}\right)\) | \(e\left(\frac{461}{638}\right)\) | \(e\left(\frac{137}{319}\right)\) |