from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8017, base_ring=CyclotomicField(4008))
M = H._module
chi = DirichletCharacter(H, M([881]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,8017))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8017\) | |
| Conductor: | \(8017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4008\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{4008})$ |
| Fixed field: | Number field defined by a degree 4008 polynomial (not computed) |
First 31 of 1328 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8017}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{613}{2004}\right)\) | \(e\left(\frac{350}{501}\right)\) | \(e\left(\frac{613}{1002}\right)\) | \(e\left(\frac{881}{4008}\right)\) | \(e\left(\frac{3}{668}\right)\) | \(e\left(\frac{269}{501}\right)\) | \(e\left(\frac{613}{668}\right)\) | \(e\left(\frac{199}{501}\right)\) | \(e\left(\frac{2107}{4008}\right)\) | \(e\left(\frac{229}{2004}\right)\) |
| \(\chi_{8017}(11,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{2004}\right)\) | \(e\left(\frac{152}{501}\right)\) | \(e\left(\frac{229}{1002}\right)\) | \(e\left(\frac{2441}{4008}\right)\) | \(e\left(\frac{279}{668}\right)\) | \(e\left(\frac{134}{501}\right)\) | \(e\left(\frac{229}{668}\right)\) | \(e\left(\frac{304}{501}\right)\) | \(e\left(\frac{2899}{4008}\right)\) | \(e\left(\frac{589}{2004}\right)\) |
| \(\chi_{8017}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1409}{2004}\right)\) | \(e\left(\frac{82}{501}\right)\) | \(e\left(\frac{407}{1002}\right)\) | \(e\left(\frac{361}{4008}\right)\) | \(e\left(\frac{579}{668}\right)\) | \(e\left(\frac{481}{501}\right)\) | \(e\left(\frac{73}{668}\right)\) | \(e\left(\frac{164}{501}\right)\) | \(e\left(\frac{3179}{4008}\right)\) | \(e\left(\frac{1445}{2004}\right)\) |
| \(\chi_{8017}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1063}{2004}\right)\) | \(e\left(\frac{128}{501}\right)\) | \(e\left(\frac{61}{1002}\right)\) | \(e\left(\frac{1871}{4008}\right)\) | \(e\left(\frac{525}{668}\right)\) | \(e\left(\frac{482}{501}\right)\) | \(e\left(\frac{395}{668}\right)\) | \(e\left(\frac{256}{501}\right)\) | \(e\left(\frac{3997}{4008}\right)\) | \(e\left(\frac{1999}{2004}\right)\) |
| \(\chi_{8017}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1235}{2004}\right)\) | \(e\left(\frac{415}{501}\right)\) | \(e\left(\frac{233}{1002}\right)\) | \(e\left(\frac{2383}{4008}\right)\) | \(e\left(\frac{297}{668}\right)\) | \(e\left(\frac{412}{501}\right)\) | \(e\left(\frac{567}{668}\right)\) | \(e\left(\frac{329}{501}\right)\) | \(e\left(\frac{845}{4008}\right)\) | \(e\left(\frac{1295}{2004}\right)\) |
| \(\chi_{8017}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{881}{2004}\right)\) | \(e\left(\frac{436}{501}\right)\) | \(e\left(\frac{881}{1002}\right)\) | \(e\left(\frac{1}{4008}\right)\) | \(e\left(\frac{207}{668}\right)\) | \(e\left(\frac{358}{501}\right)\) | \(e\left(\frac{213}{668}\right)\) | \(e\left(\frac{371}{501}\right)\) | \(e\left(\frac{1763}{4008}\right)\) | \(e\left(\frac{437}{2004}\right)\) |
| \(\chi_{8017}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{269}{2004}\right)\) | \(e\left(\frac{277}{501}\right)\) | \(e\left(\frac{269}{1002}\right)\) | \(e\left(\frac{1861}{4008}\right)\) | \(e\left(\frac{459}{668}\right)\) | \(e\left(\frac{409}{501}\right)\) | \(e\left(\frac{269}{668}\right)\) | \(e\left(\frac{53}{501}\right)\) | \(e\left(\frac{2399}{4008}\right)\) | \(e\left(\frac{1637}{2004}\right)\) |
| \(\chi_{8017}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1979}{2004}\right)\) | \(e\left(\frac{235}{501}\right)\) | \(e\left(\frac{977}{1002}\right)\) | \(e\left(\frac{3619}{4008}\right)\) | \(e\left(\frac{305}{668}\right)\) | \(e\left(\frac{16}{501}\right)\) | \(e\left(\frac{643}{668}\right)\) | \(e\left(\frac{470}{501}\right)\) | \(e\left(\frac{3569}{4008}\right)\) | \(e\left(\frac{347}{2004}\right)\) |
| \(\chi_{8017}(32,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1061}{2004}\right)\) | \(e\left(\frac{247}{501}\right)\) | \(e\left(\frac{59}{1002}\right)\) | \(e\left(\frac{397}{4008}\right)\) | \(e\left(\frac{15}{668}\right)\) | \(e\left(\frac{343}{501}\right)\) | \(e\left(\frac{393}{668}\right)\) | \(e\left(\frac{494}{501}\right)\) | \(e\left(\frac{2519}{4008}\right)\) | \(e\left(\frac{1145}{2004}\right)\) |
| \(\chi_{8017}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1085}{2004}\right)\) | \(e\left(\frac{322}{501}\right)\) | \(e\left(\frac{83}{1002}\right)\) | \(e\left(\frac{49}{4008}\right)\) | \(e\left(\frac{123}{668}\right)\) | \(e\left(\frac{7}{501}\right)\) | \(e\left(\frac{417}{668}\right)\) | \(e\left(\frac{143}{501}\right)\) | \(e\left(\frac{2219}{4008}\right)\) | \(e\left(\frac{1373}{2004}\right)\) |
| \(\chi_{8017}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{2004}\right)\) | \(e\left(\frac{284}{501}\right)\) | \(e\left(\frac{151}{1002}\right)\) | \(e\left(\frac{1067}{4008}\right)\) | \(e\left(\frac{429}{668}\right)\) | \(e\left(\frac{224}{501}\right)\) | \(e\left(\frac{151}{668}\right)\) | \(e\left(\frac{67}{501}\right)\) | \(e\left(\frac{1369}{4008}\right)\) | \(e\left(\frac{1351}{2004}\right)\) |
| \(\chi_{8017}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{805}{2004}\right)\) | \(e\left(\frac{449}{501}\right)\) | \(e\left(\frac{805}{1002}\right)\) | \(e\left(\frac{101}{4008}\right)\) | \(e\left(\frac{199}{668}\right)\) | \(e\left(\frac{86}{501}\right)\) | \(e\left(\frac{137}{668}\right)\) | \(e\left(\frac{397}{501}\right)\) | \(e\left(\frac{1711}{4008}\right)\) | \(e\left(\frac{49}{2004}\right)\) |
| \(\chi_{8017}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{911}{2004}\right)\) | \(e\left(\frac{154}{501}\right)\) | \(e\left(\frac{911}{1002}\right)\) | \(e\left(\frac{2071}{4008}\right)\) | \(e\left(\frac{509}{668}\right)\) | \(e\left(\frac{439}{501}\right)\) | \(e\left(\frac{243}{668}\right)\) | \(e\left(\frac{308}{501}\right)\) | \(e\left(\frac{3893}{4008}\right)\) | \(e\left(\frac{1223}{2004}\right)\) |
| \(\chi_{8017}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1309}{2004}\right)\) | \(e\left(\frac{20}{501}\right)\) | \(e\left(\frac{307}{1002}\right)\) | \(e\left(\frac{2813}{4008}\right)\) | \(e\left(\frac{463}{668}\right)\) | \(e\left(\frac{44}{501}\right)\) | \(e\left(\frac{641}{668}\right)\) | \(e\left(\frac{40}{501}\right)\) | \(e\left(\frac{1423}{4008}\right)\) | \(e\left(\frac{829}{2004}\right)\) |
| \(\chi_{8017}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{631}{2004}\right)\) | \(e\left(\frac{281}{501}\right)\) | \(e\left(\frac{631}{1002}\right)\) | \(e\left(\frac{2123}{4008}\right)\) | \(e\left(\frac{585}{668}\right)\) | \(e\left(\frac{17}{501}\right)\) | \(e\left(\frac{631}{668}\right)\) | \(e\left(\frac{61}{501}\right)\) | \(e\left(\frac{3385}{4008}\right)\) | \(e\left(\frac{1903}{2004}\right)\) |
| \(\chi_{8017}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{2004}\right)\) | \(e\left(\frac{302}{501}\right)\) | \(e\left(\frac{277}{1002}\right)\) | \(e\left(\frac{3749}{4008}\right)\) | \(e\left(\frac{495}{668}\right)\) | \(e\left(\frac{464}{501}\right)\) | \(e\left(\frac{277}{668}\right)\) | \(e\left(\frac{103}{501}\right)\) | \(e\left(\frac{295}{4008}\right)\) | \(e\left(\frac{1045}{2004}\right)\) |
| \(\chi_{8017}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1669}{2004}\right)\) | \(e\left(\frac{143}{501}\right)\) | \(e\left(\frac{667}{1002}\right)\) | \(e\left(\frac{1601}{4008}\right)\) | \(e\left(\frac{79}{668}\right)\) | \(e\left(\frac{14}{501}\right)\) | \(e\left(\frac{333}{668}\right)\) | \(e\left(\frac{286}{501}\right)\) | \(e\left(\frac{931}{4008}\right)\) | \(e\left(\frac{241}{2004}\right)\) |
| \(\chi_{8017}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{2004}\right)\) | \(e\left(\frac{26}{501}\right)\) | \(e\left(\frac{349}{1002}\right)\) | \(e\left(\frac{701}{4008}\right)\) | \(e\left(\frac{151}{668}\right)\) | \(e\left(\frac{458}{501}\right)\) | \(e\left(\frac{349}{668}\right)\) | \(e\left(\frac{52}{501}\right)\) | \(e\left(\frac{1399}{4008}\right)\) | \(e\left(\frac{1729}{2004}\right)\) |
| \(\chi_{8017}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1375}{2004}\right)\) | \(e\left(\frac{101}{501}\right)\) | \(e\left(\frac{373}{1002}\right)\) | \(e\left(\frac{3359}{4008}\right)\) | \(e\left(\frac{593}{668}\right)\) | \(e\left(\frac{122}{501}\right)\) | \(e\left(\frac{39}{668}\right)\) | \(e\left(\frac{202}{501}\right)\) | \(e\left(\frac{2101}{4008}\right)\) | \(e\left(\frac{955}{2004}\right)\) |
| \(\chi_{8017}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{457}{2004}\right)\) | \(e\left(\frac{113}{501}\right)\) | \(e\left(\frac{457}{1002}\right)\) | \(e\left(\frac{137}{4008}\right)\) | \(e\left(\frac{303}{668}\right)\) | \(e\left(\frac{449}{501}\right)\) | \(e\left(\frac{457}{668}\right)\) | \(e\left(\frac{226}{501}\right)\) | \(e\left(\frac{1051}{4008}\right)\) | \(e\left(\frac{1753}{2004}\right)\) |
| \(\chi_{8017}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{761}{2004}\right)\) | \(e\left(\frac{61}{501}\right)\) | \(e\left(\frac{761}{1002}\right)\) | \(e\left(\frac{3745}{4008}\right)\) | \(e\left(\frac{335}{668}\right)\) | \(e\left(\frac{34}{501}\right)\) | \(e\left(\frac{93}{668}\right)\) | \(e\left(\frac{122}{501}\right)\) | \(e\left(\frac{1259}{4008}\right)\) | \(e\left(\frac{1301}{2004}\right)\) |
| \(\chi_{8017}(99,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1025}{2004}\right)\) | \(e\left(\frac{385}{501}\right)\) | \(e\left(\frac{23}{1002}\right)\) | \(e\left(\frac{1921}{4008}\right)\) | \(e\left(\frac{187}{668}\right)\) | \(e\left(\frac{346}{501}\right)\) | \(e\left(\frac{357}{668}\right)\) | \(e\left(\frac{269}{501}\right)\) | \(e\left(\frac{3971}{4008}\right)\) | \(e\left(\frac{1805}{2004}\right)\) |
| \(\chi_{8017}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{2004}\right)\) | \(e\left(\frac{134}{501}\right)\) | \(e\left(\frac{103}{1002}\right)\) | \(e\left(\frac{1763}{4008}\right)\) | \(e\left(\frac{213}{668}\right)\) | \(e\left(\frac{395}{501}\right)\) | \(e\left(\frac{103}{668}\right)\) | \(e\left(\frac{268}{501}\right)\) | \(e\left(\frac{1969}{4008}\right)\) | \(e\left(\frac{895}{2004}\right)\) |
| \(\chi_{8017}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1495}{2004}\right)\) | \(e\left(\frac{476}{501}\right)\) | \(e\left(\frac{493}{1002}\right)\) | \(e\left(\frac{3623}{4008}\right)\) | \(e\left(\frac{465}{668}\right)\) | \(e\left(\frac{446}{501}\right)\) | \(e\left(\frac{159}{668}\right)\) | \(e\left(\frac{451}{501}\right)\) | \(e\left(\frac{2605}{4008}\right)\) | \(e\left(\frac{91}{2004}\right)\) |
| \(\chi_{8017}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{685}{2004}\right)\) | \(e\left(\frac{74}{501}\right)\) | \(e\left(\frac{685}{1002}\right)\) | \(e\left(\frac{1841}{4008}\right)\) | \(e\left(\frac{327}{668}\right)\) | \(e\left(\frac{263}{501}\right)\) | \(e\left(\frac{17}{668}\right)\) | \(e\left(\frac{148}{501}\right)\) | \(e\left(\frac{3211}{4008}\right)\) | \(e\left(\frac{913}{2004}\right)\) |
| \(\chi_{8017}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1201}{2004}\right)\) | \(e\left(\frac{434}{501}\right)\) | \(e\left(\frac{199}{1002}\right)\) | \(e\left(\frac{1373}{4008}\right)\) | \(e\left(\frac{311}{668}\right)\) | \(e\left(\frac{53}{501}\right)\) | \(e\left(\frac{533}{668}\right)\) | \(e\left(\frac{367}{501}\right)\) | \(e\left(\frac{3775}{4008}\right)\) | \(e\left(\frac{805}{2004}\right)\) |
| \(\chi_{8017}(126,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{481}{2004}\right)\) | \(e\left(\frac{188}{501}\right)\) | \(e\left(\frac{481}{1002}\right)\) | \(e\left(\frac{3797}{4008}\right)\) | \(e\left(\frac{411}{668}\right)\) | \(e\left(\frac{113}{501}\right)\) | \(e\left(\frac{481}{668}\right)\) | \(e\left(\frac{376}{501}\right)\) | \(e\left(\frac{751}{4008}\right)\) | \(e\left(\frac{1981}{2004}\right)\) |
| \(\chi_{8017}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{2004}\right)\) | \(e\left(\frac{446}{501}\right)\) | \(e\left(\frac{283}{1002}\right)\) | \(e\left(\frac{2159}{4008}\right)\) | \(e\left(\frac{21}{668}\right)\) | \(e\left(\frac{380}{501}\right)\) | \(e\left(\frac{283}{668}\right)\) | \(e\left(\frac{391}{501}\right)\) | \(e\left(\frac{2725}{4008}\right)\) | \(e\left(\frac{1603}{2004}\right)\) |
| \(\chi_{8017}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{851}{2004}\right)\) | \(e\left(\frac{217}{501}\right)\) | \(e\left(\frac{851}{1002}\right)\) | \(e\left(\frac{3943}{4008}\right)\) | \(e\left(\frac{573}{668}\right)\) | \(e\left(\frac{277}{501}\right)\) | \(e\left(\frac{183}{668}\right)\) | \(e\left(\frac{434}{501}\right)\) | \(e\left(\frac{1637}{4008}\right)\) | \(e\left(\frac{1655}{2004}\right)\) |
| \(\chi_{8017}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1889}{2004}\right)\) | \(e\left(\frac{79}{501}\right)\) | \(e\left(\frac{887}{1002}\right)\) | \(e\left(\frac{3421}{4008}\right)\) | \(e\left(\frac{67}{668}\right)\) | \(e\left(\frac{274}{501}\right)\) | \(e\left(\frac{553}{668}\right)\) | \(e\left(\frac{158}{501}\right)\) | \(e\left(\frac{3191}{4008}\right)\) | \(e\left(\frac{1997}{2004}\right)\) |
| \(\chi_{8017}(168,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{307}{2004}\right)\) | \(e\left(\frac{20}{501}\right)\) | \(e\left(\frac{307}{1002}\right)\) | \(e\left(\frac{1811}{4008}\right)\) | \(e\left(\frac{129}{668}\right)\) | \(e\left(\frac{44}{501}\right)\) | \(e\left(\frac{307}{668}\right)\) | \(e\left(\frac{40}{501}\right)\) | \(e\left(\frac{2425}{4008}\right)\) | \(e\left(\frac{1831}{2004}\right)\) |