from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243675, base_ring=CyclotomicField(1140))
M = H._module
chi = DirichletCharacter(H, M([380,513,150]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,243675))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(243675\) | |
| Conductor: | \(81225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1140\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 81225.ne | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{1140})$ |
| Fixed field: | Number field defined by a degree 1140 polynomial (not computed) |
First 31 of 288 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{243675}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1043}{1140}\right)\) | \(e\left(\frac{473}{570}\right)\) | \(e\left(\frac{73}{228}\right)\) | \(e\left(\frac{283}{380}\right)\) | \(e\left(\frac{272}{285}\right)\) | \(e\left(\frac{577}{1140}\right)\) | \(e\left(\frac{67}{285}\right)\) | \(e\left(\frac{188}{285}\right)\) | \(e\left(\frac{223}{380}\right)\) | \(e\left(\frac{991}{1140}\right)\) |
| \(\chi_{243675}(208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{901}{1140}\right)\) | \(e\left(\frac{331}{570}\right)\) | \(e\left(\frac{107}{228}\right)\) | \(e\left(\frac{141}{380}\right)\) | \(e\left(\frac{109}{285}\right)\) | \(e\left(\frac{599}{1140}\right)\) | \(e\left(\frac{74}{285}\right)\) | \(e\left(\frac{46}{285}\right)\) | \(e\left(\frac{1}{380}\right)\) | \(e\left(\frac{197}{1140}\right)\) |
| \(\chi_{243675}(1063,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{533}{1140}\right)\) | \(e\left(\frac{533}{570}\right)\) | \(e\left(\frac{163}{228}\right)\) | \(e\left(\frac{153}{380}\right)\) | \(e\left(\frac{92}{285}\right)\) | \(e\left(\frac{367}{1140}\right)\) | \(e\left(\frac{52}{285}\right)\) | \(e\left(\frac{248}{285}\right)\) | \(e\left(\frac{373}{380}\right)\) | \(e\left(\frac{901}{1140}\right)\) |
| \(\chi_{243675}(1747,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{1140}\right)\) | \(e\left(\frac{79}{570}\right)\) | \(e\left(\frac{185}{228}\right)\) | \(e\left(\frac{79}{380}\right)\) | \(e\left(\frac{181}{285}\right)\) | \(e\left(\frac{341}{1140}\right)\) | \(e\left(\frac{251}{285}\right)\) | \(e\left(\frac{79}{285}\right)\) | \(e\left(\frac{359}{380}\right)\) | \(e\left(\frac{803}{1140}\right)\) |
| \(\chi_{243675}(2602,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{1140}\right)\) | \(e\left(\frac{167}{570}\right)\) | \(e\left(\frac{13}{228}\right)\) | \(e\left(\frac{167}{380}\right)\) | \(e\left(\frac{278}{285}\right)\) | \(e\left(\frac{793}{1140}\right)\) | \(e\left(\frac{58}{285}\right)\) | \(e\left(\frac{167}{285}\right)\) | \(e\left(\frac{47}{380}\right)\) | \(e\left(\frac{139}{1140}\right)\) |
| \(\chi_{243675}(2773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{937}{1140}\right)\) | \(e\left(\frac{367}{570}\right)\) | \(e\left(\frac{47}{228}\right)\) | \(e\left(\frac{177}{380}\right)\) | \(e\left(\frac{58}{285}\right)\) | \(e\left(\frac{1043}{1140}\right)\) | \(e\left(\frac{8}{285}\right)\) | \(e\left(\frac{82}{285}\right)\) | \(e\left(\frac{357}{380}\right)\) | \(e\left(\frac{29}{1140}\right)\) |
| \(\chi_{243675}(3628,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{569}{1140}\right)\) | \(e\left(\frac{569}{570}\right)\) | \(e\left(\frac{103}{228}\right)\) | \(e\left(\frac{189}{380}\right)\) | \(e\left(\frac{41}{285}\right)\) | \(e\left(\frac{811}{1140}\right)\) | \(e\left(\frac{271}{285}\right)\) | \(e\left(\frac{284}{285}\right)\) | \(e\left(\frac{349}{380}\right)\) | \(e\left(\frac{733}{1140}\right)\) |
| \(\chi_{243675}(4312,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{343}{1140}\right)\) | \(e\left(\frac{343}{570}\right)\) | \(e\left(\frac{125}{228}\right)\) | \(e\left(\frac{343}{380}\right)\) | \(e\left(\frac{187}{285}\right)\) | \(e\left(\frac{557}{1140}\right)\) | \(e\left(\frac{242}{285}\right)\) | \(e\left(\frac{58}{285}\right)\) | \(e\left(\frac{183}{380}\right)\) | \(e\left(\frac{1091}{1140}\right)\) |
| \(\chi_{243675}(5167,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{431}{1140}\right)\) | \(e\left(\frac{431}{570}\right)\) | \(e\left(\frac{181}{228}\right)\) | \(e\left(\frac{51}{380}\right)\) | \(e\left(\frac{284}{285}\right)\) | \(e\left(\frac{1009}{1140}\right)\) | \(e\left(\frac{49}{285}\right)\) | \(e\left(\frac{146}{285}\right)\) | \(e\left(\frac{251}{380}\right)\) | \(e\left(\frac{427}{1140}\right)\) |
| \(\chi_{243675}(5338,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{973}{1140}\right)\) | \(e\left(\frac{403}{570}\right)\) | \(e\left(\frac{215}{228}\right)\) | \(e\left(\frac{213}{380}\right)\) | \(e\left(\frac{7}{285}\right)\) | \(e\left(\frac{347}{1140}\right)\) | \(e\left(\frac{227}{285}\right)\) | \(e\left(\frac{118}{285}\right)\) | \(e\left(\frac{333}{380}\right)\) | \(e\left(\frac{1001}{1140}\right)\) |
| \(\chi_{243675}(6877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{607}{1140}\right)\) | \(e\left(\frac{37}{570}\right)\) | \(e\left(\frac{65}{228}\right)\) | \(e\left(\frac{227}{380}\right)\) | \(e\left(\frac{193}{285}\right)\) | \(e\left(\frac{773}{1140}\right)\) | \(e\left(\frac{233}{285}\right)\) | \(e\left(\frac{37}{285}\right)\) | \(e\left(\frac{7}{380}\right)\) | \(e\left(\frac{239}{1140}\right)\) |
| \(\chi_{243675}(7903,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1009}{1140}\right)\) | \(e\left(\frac{439}{570}\right)\) | \(e\left(\frac{155}{228}\right)\) | \(e\left(\frac{249}{380}\right)\) | \(e\left(\frac{241}{285}\right)\) | \(e\left(\frac{791}{1140}\right)\) | \(e\left(\frac{161}{285}\right)\) | \(e\left(\frac{154}{285}\right)\) | \(e\left(\frac{309}{380}\right)\) | \(e\left(\frac{833}{1140}\right)\) |
| \(\chi_{243675}(8758,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{641}{1140}\right)\) | \(e\left(\frac{71}{570}\right)\) | \(e\left(\frac{211}{228}\right)\) | \(e\left(\frac{261}{380}\right)\) | \(e\left(\frac{224}{285}\right)\) | \(e\left(\frac{559}{1140}\right)\) | \(e\left(\frac{139}{285}\right)\) | \(e\left(\frac{71}{285}\right)\) | \(e\left(\frac{301}{380}\right)\) | \(e\left(\frac{397}{1140}\right)\) |
| \(\chi_{243675}(9442,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{871}{1140}\right)\) | \(e\left(\frac{301}{570}\right)\) | \(e\left(\frac{5}{228}\right)\) | \(e\left(\frac{111}{380}\right)\) | \(e\left(\frac{199}{285}\right)\) | \(e\left(\frac{989}{1140}\right)\) | \(e\left(\frac{224}{285}\right)\) | \(e\left(\frac{16}{285}\right)\) | \(e\left(\frac{211}{380}\right)\) | \(e\left(\frac{527}{1140}\right)\) |
| \(\chi_{243675}(10297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{959}{1140}\right)\) | \(e\left(\frac{389}{570}\right)\) | \(e\left(\frac{61}{228}\right)\) | \(e\left(\frac{199}{380}\right)\) | \(e\left(\frac{11}{285}\right)\) | \(e\left(\frac{301}{1140}\right)\) | \(e\left(\frac{31}{285}\right)\) | \(e\left(\frac{104}{285}\right)\) | \(e\left(\frac{279}{380}\right)\) | \(e\left(\frac{1003}{1140}\right)\) |
| \(\chi_{243675}(11323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{677}{1140}\right)\) | \(e\left(\frac{107}{570}\right)\) | \(e\left(\frac{151}{228}\right)\) | \(e\left(\frac{297}{380}\right)\) | \(e\left(\frac{173}{285}\right)\) | \(e\left(\frac{1003}{1140}\right)\) | \(e\left(\frac{73}{285}\right)\) | \(e\left(\frac{107}{285}\right)\) | \(e\left(\frac{277}{380}\right)\) | \(e\left(\frac{229}{1140}\right)\) |
| \(\chi_{243675}(12862,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{1140}\right)\) | \(e\left(\frac{83}{570}\right)\) | \(e\left(\frac{1}{228}\right)\) | \(e\left(\frac{83}{380}\right)\) | \(e\left(\frac{17}{285}\right)\) | \(e\left(\frac{517}{1140}\right)\) | \(e\left(\frac{22}{285}\right)\) | \(e\left(\frac{83}{285}\right)\) | \(e\left(\frac{103}{380}\right)\) | \(e\left(\frac{151}{1140}\right)\) |
| \(\chi_{243675}(13033,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1081}{1140}\right)\) | \(e\left(\frac{511}{570}\right)\) | \(e\left(\frac{35}{228}\right)\) | \(e\left(\frac{321}{380}\right)\) | \(e\left(\frac{139}{285}\right)\) | \(e\left(\frac{539}{1140}\right)\) | \(e\left(\frac{29}{285}\right)\) | \(e\left(\frac{226}{285}\right)\) | \(e\left(\frac{261}{380}\right)\) | \(e\left(\frac{497}{1140}\right)\) |
| \(\chi_{243675}(13888,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{713}{1140}\right)\) | \(e\left(\frac{143}{570}\right)\) | \(e\left(\frac{91}{228}\right)\) | \(e\left(\frac{333}{380}\right)\) | \(e\left(\frac{122}{285}\right)\) | \(e\left(\frac{307}{1140}\right)\) | \(e\left(\frac{7}{285}\right)\) | \(e\left(\frac{143}{285}\right)\) | \(e\left(\frac{253}{380}\right)\) | \(e\left(\frac{61}{1140}\right)\) |
| \(\chi_{243675}(14572,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{1140}\right)\) | \(e\left(\frac{259}{570}\right)\) | \(e\left(\frac{113}{228}\right)\) | \(e\left(\frac{259}{380}\right)\) | \(e\left(\frac{211}{285}\right)\) | \(e\left(\frac{281}{1140}\right)\) | \(e\left(\frac{206}{285}\right)\) | \(e\left(\frac{259}{285}\right)\) | \(e\left(\frac{239}{380}\right)\) | \(e\left(\frac{1103}{1140}\right)\) |
| \(\chi_{243675}(15427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{347}{1140}\right)\) | \(e\left(\frac{347}{570}\right)\) | \(e\left(\frac{169}{228}\right)\) | \(e\left(\frac{347}{380}\right)\) | \(e\left(\frac{23}{285}\right)\) | \(e\left(\frac{733}{1140}\right)\) | \(e\left(\frac{13}{285}\right)\) | \(e\left(\frac{62}{285}\right)\) | \(e\left(\frac{307}{380}\right)\) | \(e\left(\frac{439}{1140}\right)\) |
| \(\chi_{243675}(15598,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1117}{1140}\right)\) | \(e\left(\frac{547}{570}\right)\) | \(e\left(\frac{203}{228}\right)\) | \(e\left(\frac{357}{380}\right)\) | \(e\left(\frac{88}{285}\right)\) | \(e\left(\frac{983}{1140}\right)\) | \(e\left(\frac{248}{285}\right)\) | \(e\left(\frac{262}{285}\right)\) | \(e\left(\frac{237}{380}\right)\) | \(e\left(\frac{329}{1140}\right)\) |
| \(\chi_{243675}(16453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{749}{1140}\right)\) | \(e\left(\frac{179}{570}\right)\) | \(e\left(\frac{31}{228}\right)\) | \(e\left(\frac{369}{380}\right)\) | \(e\left(\frac{71}{285}\right)\) | \(e\left(\frac{751}{1140}\right)\) | \(e\left(\frac{226}{285}\right)\) | \(e\left(\frac{179}{285}\right)\) | \(e\left(\frac{229}{380}\right)\) | \(e\left(\frac{1033}{1140}\right)\) |
| \(\chi_{243675}(17137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{523}{1140}\right)\) | \(e\left(\frac{523}{570}\right)\) | \(e\left(\frac{53}{228}\right)\) | \(e\left(\frac{143}{380}\right)\) | \(e\left(\frac{217}{285}\right)\) | \(e\left(\frac{497}{1140}\right)\) | \(e\left(\frac{197}{285}\right)\) | \(e\left(\frac{238}{285}\right)\) | \(e\left(\frac{63}{380}\right)\) | \(e\left(\frac{251}{1140}\right)\) |
| \(\chi_{243675}(17992,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{611}{1140}\right)\) | \(e\left(\frac{41}{570}\right)\) | \(e\left(\frac{109}{228}\right)\) | \(e\left(\frac{231}{380}\right)\) | \(e\left(\frac{29}{285}\right)\) | \(e\left(\frac{949}{1140}\right)\) | \(e\left(\frac{4}{285}\right)\) | \(e\left(\frac{41}{285}\right)\) | \(e\left(\frac{131}{380}\right)\) | \(e\left(\frac{727}{1140}\right)\) |
| \(\chi_{243675}(18163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{1140}\right)\) | \(e\left(\frac{13}{570}\right)\) | \(e\left(\frac{143}{228}\right)\) | \(e\left(\frac{13}{380}\right)\) | \(e\left(\frac{37}{285}\right)\) | \(e\left(\frac{287}{1140}\right)\) | \(e\left(\frac{182}{285}\right)\) | \(e\left(\frac{13}{285}\right)\) | \(e\left(\frac{213}{380}\right)\) | \(e\left(\frac{161}{1140}\right)\) |
| \(\chi_{243675}(19702,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{787}{1140}\right)\) | \(e\left(\frac{217}{570}\right)\) | \(e\left(\frac{221}{228}\right)\) | \(e\left(\frac{27}{380}\right)\) | \(e\left(\frac{223}{285}\right)\) | \(e\left(\frac{713}{1140}\right)\) | \(e\left(\frac{188}{285}\right)\) | \(e\left(\frac{217}{285}\right)\) | \(e\left(\frac{267}{380}\right)\) | \(e\left(\frac{539}{1140}\right)\) |
| \(\chi_{243675}(20728,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{1140}\right)\) | \(e\left(\frac{49}{570}\right)\) | \(e\left(\frac{83}{228}\right)\) | \(e\left(\frac{49}{380}\right)\) | \(e\left(\frac{271}{285}\right)\) | \(e\left(\frac{731}{1140}\right)\) | \(e\left(\frac{116}{285}\right)\) | \(e\left(\frac{49}{285}\right)\) | \(e\left(\frac{189}{380}\right)\) | \(e\left(\frac{1133}{1140}\right)\) |
| \(\chi_{243675}(21583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{821}{1140}\right)\) | \(e\left(\frac{251}{570}\right)\) | \(e\left(\frac{139}{228}\right)\) | \(e\left(\frac{61}{380}\right)\) | \(e\left(\frac{254}{285}\right)\) | \(e\left(\frac{499}{1140}\right)\) | \(e\left(\frac{94}{285}\right)\) | \(e\left(\frac{251}{285}\right)\) | \(e\left(\frac{181}{380}\right)\) | \(e\left(\frac{697}{1140}\right)\) |
| \(\chi_{243675}(22267,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1051}{1140}\right)\) | \(e\left(\frac{481}{570}\right)\) | \(e\left(\frac{161}{228}\right)\) | \(e\left(\frac{291}{380}\right)\) | \(e\left(\frac{229}{285}\right)\) | \(e\left(\frac{929}{1140}\right)\) | \(e\left(\frac{179}{285}\right)\) | \(e\left(\frac{196}{285}\right)\) | \(e\left(\frac{91}{380}\right)\) | \(e\left(\frac{827}{1140}\right)\) |
| \(\chi_{243675}(23122,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1139}{1140}\right)\) | \(e\left(\frac{569}{570}\right)\) | \(e\left(\frac{217}{228}\right)\) | \(e\left(\frac{379}{380}\right)\) | \(e\left(\frac{41}{285}\right)\) | \(e\left(\frac{241}{1140}\right)\) | \(e\left(\frac{271}{285}\right)\) | \(e\left(\frac{284}{285}\right)\) | \(e\left(\frac{159}{380}\right)\) | \(e\left(\frac{163}{1140}\right)\) |