from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1291, base_ring=CyclotomicField(1290))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1291))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1291\) | |
| Conductor: | \(1291\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1290\) | 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: | $\Q(\zeta_{645})$ |
| Fixed field: | Number field defined by a degree 1290 polynomial (not computed) |
First 31 of 336 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1291}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1290}\right)\) | \(e\left(\frac{77}{1290}\right)\) | \(e\left(\frac{1}{645}\right)\) | \(e\left(\frac{52}{215}\right)\) | \(e\left(\frac{13}{215}\right)\) | \(e\left(\frac{266}{645}\right)\) | \(e\left(\frac{1}{430}\right)\) | \(e\left(\frac{77}{645}\right)\) | \(e\left(\frac{313}{1290}\right)\) | \(e\left(\frac{193}{258}\right)\) |
| \(\chi_{1291}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{1290}\right)\) | \(e\left(\frac{769}{1290}\right)\) | \(e\left(\frac{77}{645}\right)\) | \(e\left(\frac{134}{215}\right)\) | \(e\left(\frac{141}{215}\right)\) | \(e\left(\frac{487}{645}\right)\) | \(e\left(\frac{77}{430}\right)\) | \(e\left(\frac{124}{645}\right)\) | \(e\left(\frac{881}{1290}\right)\) | \(e\left(\frac{155}{258}\right)\) |
| \(\chi_{1291}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{1290}\right)\) | \(e\left(\frac{881}{1290}\right)\) | \(e\left(\frac{313}{645}\right)\) | \(e\left(\frac{151}{215}\right)\) | \(e\left(\frac{199}{215}\right)\) | \(e\left(\frac{53}{645}\right)\) | \(e\left(\frac{313}{430}\right)\) | \(e\left(\frac{236}{645}\right)\) | \(e\left(\frac{1219}{1290}\right)\) | \(e\left(\frac{37}{258}\right)\) |
| \(\chi_{1291}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{1290}\right)\) | \(e\left(\frac{923}{1290}\right)\) | \(e\left(\frac{79}{645}\right)\) | \(e\left(\frac{23}{215}\right)\) | \(e\left(\frac{167}{215}\right)\) | \(e\left(\frac{374}{645}\right)\) | \(e\left(\frac{79}{430}\right)\) | \(e\left(\frac{278}{645}\right)\) | \(e\left(\frac{217}{1290}\right)\) | \(e\left(\frac{25}{258}\right)\) |
| \(\chi_{1291}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{533}{1290}\right)\) | \(e\left(\frac{1051}{1290}\right)\) | \(e\left(\frac{533}{645}\right)\) | \(e\left(\frac{196}{215}\right)\) | \(e\left(\frac{49}{215}\right)\) | \(e\left(\frac{523}{645}\right)\) | \(e\left(\frac{103}{430}\right)\) | \(e\left(\frac{406}{645}\right)\) | \(e\left(\frac{419}{1290}\right)\) | \(e\left(\frac{185}{258}\right)\) |
| \(\chi_{1291}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{389}{1290}\right)\) | \(e\left(\frac{283}{1290}\right)\) | \(e\left(\frac{389}{645}\right)\) | \(e\left(\frac{18}{215}\right)\) | \(e\left(\frac{112}{215}\right)\) | \(e\left(\frac{274}{645}\right)\) | \(e\left(\frac{389}{430}\right)\) | \(e\left(\frac{283}{645}\right)\) | \(e\left(\frac{497}{1290}\right)\) | \(e\left(\frac{257}{258}\right)\) |
| \(\chi_{1291}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{1290}\right)\) | \(e\left(\frac{209}{1290}\right)\) | \(e\left(\frac{187}{645}\right)\) | \(e\left(\frac{49}{215}\right)\) | \(e\left(\frac{66}{215}\right)\) | \(e\left(\frac{77}{645}\right)\) | \(e\left(\frac{187}{430}\right)\) | \(e\left(\frac{209}{645}\right)\) | \(e\left(\frac{481}{1290}\right)\) | \(e\left(\frac{229}{258}\right)\) |
| \(\chi_{1291}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{1290}\right)\) | \(e\left(\frac{1289}{1290}\right)\) | \(e\left(\frac{67}{645}\right)\) | \(e\left(\frac{44}{215}\right)\) | \(e\left(\frac{11}{215}\right)\) | \(e\left(\frac{407}{645}\right)\) | \(e\left(\frac{67}{430}\right)\) | \(e\left(\frac{644}{645}\right)\) | \(e\left(\frac{331}{1290}\right)\) | \(e\left(\frac{31}{258}\right)\) |
| \(\chi_{1291}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{967}{1290}\right)\) | \(e\left(\frac{929}{1290}\right)\) | \(e\left(\frac{322}{645}\right)\) | \(e\left(\frac{189}{215}\right)\) | \(e\left(\frac{101}{215}\right)\) | \(e\left(\frac{512}{645}\right)\) | \(e\left(\frac{107}{430}\right)\) | \(e\left(\frac{284}{645}\right)\) | \(e\left(\frac{811}{1290}\right)\) | \(e\left(\frac{97}{258}\right)\) |
| \(\chi_{1291}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1277}{1290}\right)\) | \(e\left(\frac{289}{1290}\right)\) | \(e\left(\frac{632}{645}\right)\) | \(e\left(\frac{184}{215}\right)\) | \(e\left(\frac{46}{215}\right)\) | \(e\left(\frac{412}{645}\right)\) | \(e\left(\frac{417}{430}\right)\) | \(e\left(\frac{289}{645}\right)\) | \(e\left(\frac{1091}{1290}\right)\) | \(e\left(\frac{71}{258}\right)\) |
| \(\chi_{1291}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{1290}\right)\) | \(e\left(\frac{749}{1290}\right)\) | \(e\left(\frac{127}{645}\right)\) | \(e\left(\frac{154}{215}\right)\) | \(e\left(\frac{146}{215}\right)\) | \(e\left(\frac{242}{645}\right)\) | \(e\left(\frac{127}{430}\right)\) | \(e\left(\frac{104}{645}\right)\) | \(e\left(\frac{1051}{1290}\right)\) | \(e\left(\frac{1}{258}\right)\) |
| \(\chi_{1291}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{391}{1290}\right)\) | \(e\left(\frac{437}{1290}\right)\) | \(e\left(\frac{391}{645}\right)\) | \(e\left(\frac{122}{215}\right)\) | \(e\left(\frac{138}{215}\right)\) | \(e\left(\frac{161}{645}\right)\) | \(e\left(\frac{391}{430}\right)\) | \(e\left(\frac{437}{645}\right)\) | \(e\left(\frac{1123}{1290}\right)\) | \(e\left(\frac{127}{258}\right)\) |
| \(\chi_{1291}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{349}{1290}\right)\) | \(e\left(\frac{1073}{1290}\right)\) | \(e\left(\frac{349}{645}\right)\) | \(e\left(\frac{88}{215}\right)\) | \(e\left(\frac{22}{215}\right)\) | \(e\left(\frac{599}{645}\right)\) | \(e\left(\frac{349}{430}\right)\) | \(e\left(\frac{428}{645}\right)\) | \(e\left(\frac{877}{1290}\right)\) | \(e\left(\frac{19}{258}\right)\) |
| \(\chi_{1291}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1043}{1290}\right)\) | \(e\left(\frac{331}{1290}\right)\) | \(e\left(\frac{398}{645}\right)\) | \(e\left(\frac{56}{215}\right)\) | \(e\left(\frac{14}{215}\right)\) | \(e\left(\frac{88}{645}\right)\) | \(e\left(\frac{183}{430}\right)\) | \(e\left(\frac{331}{645}\right)\) | \(e\left(\frac{89}{1290}\right)\) | \(e\left(\frac{59}{258}\right)\) |
| \(\chi_{1291}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{1290}\right)\) | \(e\left(\frac{479}{1290}\right)\) | \(e\left(\frac{157}{645}\right)\) | \(e\left(\frac{209}{215}\right)\) | \(e\left(\frac{106}{215}\right)\) | \(e\left(\frac{482}{645}\right)\) | \(e\left(\frac{157}{430}\right)\) | \(e\left(\frac{479}{645}\right)\) | \(e\left(\frac{121}{1290}\right)\) | \(e\left(\frac{115}{258}\right)\) |
| \(\chi_{1291}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{449}{1290}\right)\) | \(e\left(\frac{1033}{1290}\right)\) | \(e\left(\frac{449}{645}\right)\) | \(e\left(\frac{128}{215}\right)\) | \(e\left(\frac{32}{215}\right)\) | \(e\left(\frac{109}{645}\right)\) | \(e\left(\frac{19}{430}\right)\) | \(e\left(\frac{388}{645}\right)\) | \(e\left(\frac{1217}{1290}\right)\) | \(e\left(\frac{227}{258}\right)\) |
| \(\chi_{1291}(75,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{701}{1290}\right)\) | \(e\left(\frac{1087}{1290}\right)\) | \(e\left(\frac{56}{645}\right)\) | \(e\left(\frac{117}{215}\right)\) | \(e\left(\frac{83}{215}\right)\) | \(e\left(\frac{61}{645}\right)\) | \(e\left(\frac{271}{430}\right)\) | \(e\left(\frac{442}{645}\right)\) | \(e\left(\frac{113}{1290}\right)\) | \(e\left(\frac{101}{258}\right)\) |
| \(\chi_{1291}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{611}{1290}\right)\) | \(e\left(\frac{607}{1290}\right)\) | \(e\left(\frac{611}{645}\right)\) | \(e\left(\frac{167}{215}\right)\) | \(e\left(\frac{203}{215}\right)\) | \(e\left(\frac{631}{645}\right)\) | \(e\left(\frac{181}{430}\right)\) | \(e\left(\frac{607}{645}\right)\) | \(e\left(\frac{323}{1290}\right)\) | \(e\left(\frac{17}{258}\right)\) |
| \(\chi_{1291}(90,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{467}{1290}\right)\) | \(e\left(\frac{1129}{1290}\right)\) | \(e\left(\frac{467}{645}\right)\) | \(e\left(\frac{204}{215}\right)\) | \(e\left(\frac{51}{215}\right)\) | \(e\left(\frac{382}{645}\right)\) | \(e\left(\frac{37}{430}\right)\) | \(e\left(\frac{484}{645}\right)\) | \(e\left(\frac{401}{1290}\right)\) | \(e\left(\frac{89}{258}\right)\) |
| \(\chi_{1291}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{1290}\right)\) | \(e\left(\frac{557}{1290}\right)\) | \(e\left(\frac{91}{645}\right)\) | \(e\left(\frac{2}{215}\right)\) | \(e\left(\frac{108}{215}\right)\) | \(e\left(\frac{341}{645}\right)\) | \(e\left(\frac{91}{430}\right)\) | \(e\left(\frac{557}{645}\right)\) | \(e\left(\frac{103}{1290}\right)\) | \(e\left(\frac{19}{258}\right)\) |
| \(\chi_{1291}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{1290}\right)\) | \(e\left(\frac{1229}{1290}\right)\) | \(e\left(\frac{217}{645}\right)\) | \(e\left(\frac{104}{215}\right)\) | \(e\left(\frac{26}{215}\right)\) | \(e\left(\frac{317}{645}\right)\) | \(e\left(\frac{217}{430}\right)\) | \(e\left(\frac{584}{645}\right)\) | \(e\left(\frac{841}{1290}\right)\) | \(e\left(\frac{85}{258}\right)\) |
| \(\chi_{1291}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{553}{1290}\right)\) | \(e\left(\frac{11}{1290}\right)\) | \(e\left(\frac{553}{645}\right)\) | \(e\left(\frac{161}{215}\right)\) | \(e\left(\frac{94}{215}\right)\) | \(e\left(\frac{38}{645}\right)\) | \(e\left(\frac{123}{430}\right)\) | \(e\left(\frac{11}{645}\right)\) | \(e\left(\frac{229}{1290}\right)\) | \(e\left(\frac{175}{258}\right)\) |
| \(\chi_{1291}(104,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{821}{1290}\right)\) | \(e\left(\frac{7}{1290}\right)\) | \(e\left(\frac{176}{645}\right)\) | \(e\left(\frac{122}{215}\right)\) | \(e\left(\frac{138}{215}\right)\) | \(e\left(\frac{376}{645}\right)\) | \(e\left(\frac{391}{430}\right)\) | \(e\left(\frac{7}{645}\right)\) | \(e\left(\frac{263}{1290}\right)\) | \(e\left(\frac{41}{258}\right)\) |
| \(\chi_{1291}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{1290}\right)\) | \(e\left(\frac{1171}{1290}\right)\) | \(e\left(\frac{233}{645}\right)\) | \(e\left(\frac{76}{215}\right)\) | \(e\left(\frac{19}{215}\right)\) | \(e\left(\frac{58}{645}\right)\) | \(e\left(\frac{233}{430}\right)\) | \(e\left(\frac{526}{645}\right)\) | \(e\left(\frac{689}{1290}\right)\) | \(e\left(\frac{77}{258}\right)\) |
| \(\chi_{1291}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1159}{1290}\right)\) | \(e\left(\frac{233}{1290}\right)\) | \(e\left(\frac{514}{645}\right)\) | \(e\left(\frac{68}{215}\right)\) | \(e\left(\frac{17}{215}\right)\) | \(e\left(\frac{629}{645}\right)\) | \(e\left(\frac{299}{430}\right)\) | \(e\left(\frac{233}{645}\right)\) | \(e\left(\frac{277}{1290}\right)\) | \(e\left(\frac{1}{258}\right)\) |
| \(\chi_{1291}(113,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1051}{1290}\right)\) | \(e\left(\frac{947}{1290}\right)\) | \(e\left(\frac{406}{645}\right)\) | \(e\left(\frac{42}{215}\right)\) | \(e\left(\frac{118}{215}\right)\) | \(e\left(\frac{281}{645}\right)\) | \(e\left(\frac{191}{430}\right)\) | \(e\left(\frac{302}{645}\right)\) | \(e\left(\frac{13}{1290}\right)\) | \(e\left(\frac{55}{258}\right)\) |
| \(\chi_{1291}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{499}{1290}\right)\) | \(e\left(\frac{1013}{1290}\right)\) | \(e\left(\frac{499}{645}\right)\) | \(e\left(\frac{148}{215}\right)\) | \(e\left(\frac{37}{215}\right)\) | \(e\left(\frac{509}{645}\right)\) | \(e\left(\frac{69}{430}\right)\) | \(e\left(\frac{368}{645}\right)\) | \(e\left(\frac{97}{1290}\right)\) | \(e\left(\frac{73}{258}\right)\) |
| \(\chi_{1291}(118,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{857}{1290}\right)\) | \(e\left(\frac{199}{1290}\right)\) | \(e\left(\frac{212}{645}\right)\) | \(e\left(\frac{59}{215}\right)\) | \(e\left(\frac{176}{215}\right)\) | \(e\left(\frac{277}{645}\right)\) | \(e\left(\frac{427}{430}\right)\) | \(e\left(\frac{199}{645}\right)\) | \(e\left(\frac{1211}{1290}\right)\) | \(e\left(\frac{23}{258}\right)\) |
| \(\chi_{1291}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{941}{1290}\right)\) | \(e\left(\frac{217}{1290}\right)\) | \(e\left(\frac{296}{645}\right)\) | \(e\left(\frac{127}{215}\right)\) | \(e\left(\frac{193}{215}\right)\) | \(e\left(\frac{46}{645}\right)\) | \(e\left(\frac{81}{430}\right)\) | \(e\left(\frac{217}{645}\right)\) | \(e\left(\frac{413}{1290}\right)\) | \(e\left(\frac{239}{258}\right)\) |
| \(\chi_{1291}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{1290}\right)\) | \(e\left(\frac{539}{1290}\right)\) | \(e\left(\frac{7}{645}\right)\) | \(e\left(\frac{149}{215}\right)\) | \(e\left(\frac{91}{215}\right)\) | \(e\left(\frac{572}{645}\right)\) | \(e\left(\frac{7}{430}\right)\) | \(e\left(\frac{539}{645}\right)\) | \(e\left(\frac{901}{1290}\right)\) | \(e\left(\frac{61}{258}\right)\) |
| \(\chi_{1291}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1027}{1290}\right)\) | \(e\left(\frac{389}{1290}\right)\) | \(e\left(\frac{382}{645}\right)\) | \(e\left(\frac{84}{215}\right)\) | \(e\left(\frac{21}{215}\right)\) | \(e\left(\frac{347}{645}\right)\) | \(e\left(\frac{167}{430}\right)\) | \(e\left(\frac{389}{645}\right)\) | \(e\left(\frac{241}{1290}\right)\) | \(e\left(\frac{67}{258}\right)\) |