from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987696, base_ring=CyclotomicField(2166))
M = H._module
chi = DirichletCharacter(H, M([0,0,361,1327]))
chi.galois_orbit()
[g,chi] = znchar(Mod(65,987696))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(987696\) | |
| Conductor: | \(61731\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2166\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 61731.cz | 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: | $\Q(\zeta_{1083})$ |
| Fixed field: | Number field defined by a degree 2166 polynomial (not computed) |
First 31 of 684 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{987696}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{243}{722}\right)\) | \(e\left(\frac{98}{1083}\right)\) | \(e\left(\frac{1309}{2166}\right)\) | \(e\left(\frac{1711}{2166}\right)\) | \(e\left(\frac{937}{2166}\right)\) | \(e\left(\frac{151}{2166}\right)\) | \(e\left(\frac{243}{361}\right)\) | \(e\left(\frac{77}{361}\right)\) | \(e\left(\frac{611}{2166}\right)\) | \(e\left(\frac{925}{2166}\right)\) |
| \(\chi_{987696}(2273,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{201}{722}\right)\) | \(e\left(\frac{139}{1083}\right)\) | \(e\left(\frac{851}{2166}\right)\) | \(e\left(\frac{515}{2166}\right)\) | \(e\left(\frac{677}{2166}\right)\) | \(e\left(\frac{1043}{2166}\right)\) | \(e\left(\frac{201}{361}\right)\) | \(e\left(\frac{135}{361}\right)\) | \(e\left(\frac{1165}{2166}\right)\) | \(e\left(\frac{881}{2166}\right)\) |
| \(\chi_{987696}(2801,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{722}\right)\) | \(e\left(\frac{632}{1083}\right)\) | \(e\left(\frac{1789}{2166}\right)\) | \(e\left(\frac{1243}{2166}\right)\) | \(e\left(\frac{1777}{2166}\right)\) | \(e\left(\frac{2101}{2166}\right)\) | \(e\left(\frac{101}{361}\right)\) | \(e\left(\frac{84}{361}\right)\) | \(e\left(\frac{1487}{2166}\right)\) | \(e\left(\frac{1567}{2166}\right)\) |
| \(\chi_{987696}(5009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{461}{722}\right)\) | \(e\left(\frac{229}{1083}\right)\) | \(e\left(\frac{1589}{2166}\right)\) | \(e\left(\frac{1799}{2166}\right)\) | \(e\left(\frac{1427}{2166}\right)\) | \(e\left(\frac{1469}{2166}\right)\) | \(e\left(\frac{100}{361}\right)\) | \(e\left(\frac{51}{361}\right)\) | \(e\left(\frac{1483}{2166}\right)\) | \(e\left(\frac{1841}{2166}\right)\) |
| \(\chi_{987696}(5537,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{722}\right)\) | \(e\left(\frac{482}{1083}\right)\) | \(e\left(\frac{559}{2166}\right)\) | \(e\left(\frac{547}{2166}\right)\) | \(e\left(\frac{1249}{2166}\right)\) | \(e\left(\frac{2113}{2166}\right)\) | \(e\left(\frac{149}{361}\right)\) | \(e\left(\frac{224}{361}\right)\) | \(e\left(\frac{1679}{2166}\right)\) | \(e\left(\frac{1411}{2166}\right)\) |
| \(\chi_{987696}(7745,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{645}{722}\right)\) | \(e\left(\frac{376}{1083}\right)\) | \(e\left(\frac{845}{2166}\right)\) | \(e\left(\frac{575}{2166}\right)\) | \(e\left(\frac{125}{2166}\right)\) | \(e\left(\frac{71}{2166}\right)\) | \(e\left(\frac{284}{361}\right)\) | \(e\left(\frac{347}{361}\right)\) | \(e\left(\frac{775}{2166}\right)\) | \(e\left(\frac{521}{2166}\right)\) |
| \(\chi_{987696}(8273,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{387}{722}\right)\) | \(e\left(\frac{731}{1083}\right)\) | \(e\left(\frac{1951}{2166}\right)\) | \(e\left(\frac{1789}{2166}\right)\) | \(e\left(\frac{1519}{2166}\right)\) | \(e\left(\frac{187}{2166}\right)\) | \(e\left(\frac{26}{361}\right)\) | \(e\left(\frac{136}{361}\right)\) | \(e\left(\frac{1187}{2166}\right)\) | \(e\left(\frac{457}{2166}\right)\) |
| \(\chi_{987696}(10481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{722}\right)\) | \(e\left(\frac{580}{1083}\right)\) | \(e\left(\frac{785}{2166}\right)\) | \(e\left(\frac{1175}{2166}\right)\) | \(e\left(\frac{1103}{2166}\right)\) | \(e\left(\frac{1181}{2166}\right)\) | \(e\left(\frac{31}{361}\right)\) | \(e\left(\frac{301}{361}\right)\) | \(e\left(\frac{1207}{2166}\right)\) | \(e\left(\frac{1253}{2166}\right)\) |
| \(\chi_{987696}(11009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{722}\right)\) | \(e\left(\frac{296}{1083}\right)\) | \(e\left(\frac{1633}{2166}\right)\) | \(e\left(\frac{637}{2166}\right)\) | \(e\left(\frac{421}{2166}\right)\) | \(e\left(\frac{655}{2166}\right)\) | \(e\left(\frac{93}{361}\right)\) | \(e\left(\frac{181}{361}\right)\) | \(e\left(\frac{11}{2166}\right)\) | \(e\left(\frac{871}{2166}\right)\) |
| \(\chi_{987696}(13217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{722}\right)\) | \(e\left(\frac{841}{1083}\right)\) | \(e\left(\frac{1409}{2166}\right)\) | \(e\left(\frac{1433}{2166}\right)\) | \(e\left(\frac{29}{2166}\right)\) | \(e\left(\frac{467}{2166}\right)\) | \(e\left(\frac{63}{361}\right)\) | \(e\left(\frac{274}{361}\right)\) | \(e\left(\frac{613}{2166}\right)\) | \(e\left(\frac{1871}{2166}\right)\) |
| \(\chi_{987696}(13745,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{711}{722}\right)\) | \(e\left(\frac{260}{1083}\right)\) | \(e\left(\frac{1771}{2166}\right)\) | \(e\left(\frac{1423}{2166}\right)\) | \(e\left(\frac{121}{2166}\right)\) | \(e\left(\frac{1351}{2166}\right)\) | \(e\left(\frac{350}{361}\right)\) | \(e\left(\frac{359}{361}\right)\) | \(e\left(\frac{317}{2166}\right)\) | \(e\left(\frac{487}{2166}\right)\) |
| \(\chi_{987696}(16481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{722}\right)\) | \(e\left(\frac{623}{1083}\right)\) | \(e\left(\frac{199}{2166}\right)\) | \(e\left(\frac{1981}{2166}\right)\) | \(e\left(\frac{619}{2166}\right)\) | \(e\left(\frac{109}{2166}\right)\) | \(e\left(\frac{75}{361}\right)\) | \(e\left(\frac{309}{361}\right)\) | \(e\left(\frac{2105}{2166}\right)\) | \(e\left(\frac{1471}{2166}\right)\) |
| \(\chi_{987696}(18689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{621}{722}\right)\) | \(e\left(\frac{451}{1083}\right)\) | \(e\left(\frac{377}{2166}\right)\) | \(e\left(\frac{923}{2166}\right)\) | \(e\left(\frac{389}{2166}\right)\) | \(e\left(\frac{65}{2166}\right)\) | \(e\left(\frac{260}{361}\right)\) | \(e\left(\frac{277}{361}\right)\) | \(e\left(\frac{679}{2166}\right)\) | \(e\left(\frac{599}{2166}\right)\) |
| \(\chi_{987696}(19217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{351}{722}\right)\) | \(e\left(\frac{302}{1083}\right)\) | \(e\left(\frac{1249}{2166}\right)\) | \(e\left(\frac{145}{2166}\right)\) | \(e\left(\frac{1915}{2166}\right)\) | \(e\left(\frac{1261}{2166}\right)\) | \(e\left(\frac{351}{361}\right)\) | \(e\left(\frac{31}{361}\right)\) | \(e\left(\frac{1043}{2166}\right)\) | \(e\left(\frac{1657}{2166}\right)\) |
| \(\chi_{987696}(21425,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{425}{722}\right)\) | \(e\left(\frac{883}{1083}\right)\) | \(e\left(\frac{887}{2166}\right)\) | \(e\left(\frac{155}{2166}\right)\) | \(e\left(\frac{1823}{2166}\right)\) | \(e\left(\frac{377}{2166}\right)\) | \(e\left(\frac{64}{361}\right)\) | \(e\left(\frac{307}{361}\right)\) | \(e\left(\frac{1339}{2166}\right)\) | \(e\left(\frac{875}{2166}\right)\) |
| \(\chi_{987696}(24161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{722}\right)\) | \(e\left(\frac{289}{1083}\right)\) | \(e\left(\frac{2081}{2166}\right)\) | \(e\left(\frac{1211}{2166}\right)\) | \(e\left(\frac{1205}{2166}\right)\) | \(e\left(\frac{1031}{2166}\right)\) | \(e\left(\frac{153}{361}\right)\) | \(e\left(\frac{356}{361}\right)\) | \(e\left(\frac{973}{2166}\right)\) | \(e\left(\frac{1037}{2166}\right)\) |
| \(\chi_{987696}(24689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{722}\right)\) | \(e\left(\frac{857}{1083}\right)\) | \(e\left(\frac{385}{2166}\right)\) | \(e\left(\frac{121}{2166}\right)\) | \(e\left(\frac{403}{2166}\right)\) | \(e\left(\frac{2083}{2166}\right)\) | \(e\left(\frac{29}{361}\right)\) | \(e\left(\frac{235}{361}\right)\) | \(e\left(\frac{1199}{2166}\right)\) | \(e\left(\frac{1801}{2166}\right)\) |
| \(\chi_{987696}(26897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{527}{722}\right)\) | \(e\left(\frac{835}{1083}\right)\) | \(e\left(\frac{1793}{2166}\right)\) | \(e\left(\frac{1925}{2166}\right)\) | \(e\left(\frac{701}{2166}\right)\) | \(e\left(\frac{2027}{2166}\right)\) | \(e\left(\frac{166}{361}\right)\) | \(e\left(\frac{63}{361}\right)\) | \(e\left(\frac{1747}{2166}\right)\) | \(e\left(\frac{1085}{2166}\right)\) |
| \(\chi_{987696}(27425,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{722}\right)\) | \(e\left(\frac{650}{1083}\right)\) | \(e\left(\frac{637}{2166}\right)\) | \(e\left(\frac{1933}{2166}\right)\) | \(e\left(\frac{1927}{2166}\right)\) | \(e\left(\frac{1753}{2166}\right)\) | \(e\left(\frac{153}{361}\right)\) | \(e\left(\frac{356}{361}\right)\) | \(e\left(\frac{251}{2166}\right)\) | \(e\left(\frac{1759}{2166}\right)\) |
| \(\chi_{987696}(29633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{722}\right)\) | \(e\left(\frac{355}{1083}\right)\) | \(e\left(\frac{23}{2166}\right)\) | \(e\left(\frac{131}{2166}\right)\) | \(e\left(\frac{311}{2166}\right)\) | \(e\left(\frac{1199}{2166}\right)\) | \(e\left(\frac{103}{361}\right)\) | \(e\left(\frac{150}{361}\right)\) | \(e\left(\frac{1495}{2166}\right)\) | \(e\left(\frac{1019}{2166}\right)\) |
| \(\chi_{987696}(30161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{467}{722}\right)\) | \(e\left(\frac{842}{1083}\right)\) | \(e\left(\frac{1345}{2166}\right)\) | \(e\left(\frac{1351}{2166}\right)\) | \(e\left(\frac{2083}{2166}\right)\) | \(e\left(\frac{1651}{2166}\right)\) | \(e\left(\frac{106}{361}\right)\) | \(e\left(\frac{249}{361}\right)\) | \(e\left(\frac{785}{2166}\right)\) | \(e\left(\frac{919}{2166}\right)\) |
| \(\chi_{987696}(32369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{325}{722}\right)\) | \(e\left(\frac{1015}{1083}\right)\) | \(e\left(\frac{1103}{2166}\right)\) | \(e\left(\frac{161}{2166}\right)\) | \(e\left(\frac{35}{2166}\right)\) | \(e\left(\frac{713}{2166}\right)\) | \(e\left(\frac{325}{361}\right)\) | \(e\left(\frac{256}{361}\right)\) | \(e\left(\frac{217}{2166}\right)\) | \(e\left(\frac{839}{2166}\right)\) |
| \(\chi_{987696}(32897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{249}{722}\right)\) | \(e\left(\frac{350}{1083}\right)\) | \(e\left(\frac{343}{2166}\right)\) | \(e\left(\frac{541}{2166}\right)\) | \(e\left(\frac{871}{2166}\right)\) | \(e\left(\frac{1777}{2166}\right)\) | \(e\left(\frac{249}{361}\right)\) | \(e\left(\frac{275}{361}\right)\) | \(e\left(\frac{635}{2166}\right)\) | \(e\left(\frac{1447}{2166}\right)\) |
| \(\chi_{987696}(35105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{471}{722}\right)\) | \(e\left(\frac{649}{1083}\right)\) | \(e\left(\frac{701}{2166}\right)\) | \(e\left(\frac{2015}{2166}\right)\) | \(e\left(\frac{2039}{2166}\right)\) | \(e\left(\frac{569}{2166}\right)\) | \(e\left(\frac{110}{361}\right)\) | \(e\left(\frac{20}{361}\right)\) | \(e\left(\frac{79}{2166}\right)\) | \(e\left(\frac{545}{2166}\right)\) |
| \(\chi_{987696}(35633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{221}{722}\right)\) | \(e\left(\frac{257}{1083}\right)\) | \(e\left(\frac{1963}{2166}\right)\) | \(e\left(\frac{1669}{2166}\right)\) | \(e\left(\frac{457}{2166}\right)\) | \(e\left(\frac{2131}{2166}\right)\) | \(e\left(\frac{221}{361}\right)\) | \(e\left(\frac{73}{361}\right)\) | \(e\left(\frac{1967}{2166}\right)\) | \(e\left(\frac{1177}{2166}\right)\) |
| \(\chi_{987696}(37841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{541}{722}\right)\) | \(e\left(\frac{340}{1083}\right)\) | \(e\left(\frac{983}{2166}\right)\) | \(e\left(\frac{1361}{2166}\right)\) | \(e\left(\frac{1991}{2166}\right)\) | \(e\left(\frac{767}{2166}\right)\) | \(e\left(\frac{180}{361}\right)\) | \(e\left(\frac{164}{361}\right)\) | \(e\left(\frac{1081}{2166}\right)\) | \(e\left(\frac{137}{2166}\right)\) |
| \(\chi_{987696}(38369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{722}\right)\) | \(e\left(\frac{563}{1083}\right)\) | \(e\left(\frac{1873}{2166}\right)\) | \(e\left(\frac{403}{2166}\right)\) | \(e\left(\frac{841}{2166}\right)\) | \(e\left(\frac{547}{2166}\right)\) | \(e\left(\frac{22}{361}\right)\) | \(e\left(\frac{4}{361}\right)\) | \(e\left(\frac{449}{2166}\right)\) | \(e\left(\frac{109}{2166}\right)\) |
| \(\chi_{987696}(40577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{535}{722}\right)\) | \(e\left(\frac{88}{1083}\right)\) | \(e\left(\frac{1949}{2166}\right)\) | \(e\left(\frac{365}{2166}\right)\) | \(e\left(\frac{2057}{2166}\right)\) | \(e\left(\frac{1307}{2166}\right)\) | \(e\left(\frac{174}{361}\right)\) | \(e\left(\frac{327}{361}\right)\) | \(e\left(\frac{1057}{2166}\right)\) | \(e\left(\frac{1781}{2166}\right)\) |
| \(\chi_{987696}(41105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{722}\right)\) | \(e\left(\frac{185}{1083}\right)\) | \(e\left(\frac{73}{2166}\right)\) | \(e\left(\frac{1075}{2166}\right)\) | \(e\left(\frac{2023}{2166}\right)\) | \(e\left(\frac{1357}{2166}\right)\) | \(e\left(\frac{13}{361}\right)\) | \(e\left(\frac{68}{361}\right)\) | \(e\left(\frac{413}{2166}\right)\) | \(e\left(\frac{409}{2166}\right)\) |
| \(\chi_{987696}(43313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{453}{722}\right)\) | \(e\left(\frac{976}{1083}\right)\) | \(e\left(\frac{1433}{2166}\right)\) | \(e\left(\frac{1193}{2166}\right)\) | \(e\left(\frac{71}{2166}\right)\) | \(e\left(\frac{23}{2166}\right)\) | \(e\left(\frac{92}{361}\right)\) | \(e\left(\frac{148}{361}\right)\) | \(e\left(\frac{7}{2166}\right)\) | \(e\left(\frac{1145}{2166}\right)\) |
| \(\chi_{987696}(43841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{555}{722}\right)\) | \(e\left(\frac{206}{1083}\right)\) | \(e\left(\frac{895}{2166}\right)\) | \(e\left(\frac{1519}{2166}\right)\) | \(e\left(\frac{1837}{2166}\right)\) | \(e\left(\frac{229}{2166}\right)\) | \(e\left(\frac{194}{361}\right)\) | \(e\left(\frac{265}{361}\right)\) | \(e\left(\frac{1859}{2166}\right)\) | \(e\left(\frac{2077}{2166}\right)\) |