from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4019, base_ring=CyclotomicField(4018))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,4019))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4019\) | |
| Conductor: | \(4019\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2009\) | 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: | $\Q(\zeta_{2009})$ |
| Fixed field: | Number field defined by a degree 2009 polynomial (not computed) |
First 31 of 1680 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4019}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{2009}\right)\) | \(e\left(\frac{25}{49}\right)\) | \(e\left(\frac{2}{2009}\right)\) | \(e\left(\frac{796}{2009}\right)\) | \(e\left(\frac{1026}{2009}\right)\) | \(e\left(\frac{1760}{2009}\right)\) | \(e\left(\frac{3}{2009}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{797}{2009}\right)\) | \(e\left(\frac{261}{2009}\right)\) |
| \(\chi_{4019}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{398}{2009}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{796}{2009}\right)\) | \(e\left(\frac{1395}{2009}\right)\) | \(e\left(\frac{521}{2009}\right)\) | \(e\left(\frac{1348}{2009}\right)\) | \(e\left(\frac{1194}{2009}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{1793}{2009}\right)\) | \(e\left(\frac{1419}{2009}\right)\) |
| \(\chi_{4019}(12,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1518}{2009}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{1027}{2009}\right)\) | \(e\left(\frac{919}{2009}\right)\) | \(e\left(\frac{493}{2009}\right)\) | \(e\left(\frac{1719}{2009}\right)\) | \(e\left(\frac{536}{2009}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{428}{2009}\right)\) | \(e\left(\frac{425}{2009}\right)\) |
| \(\chi_{4019}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1885}{2009}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{1761}{2009}\right)\) | \(e\left(\frac{1746}{2009}\right)\) | \(e\left(\frac{1352}{2009}\right)\) | \(e\left(\frac{741}{2009}\right)\) | \(e\left(\frac{1637}{2009}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{1622}{2009}\right)\) | \(e\left(\frac{1789}{2009}\right)\) |
| \(\chi_{4019}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1915}{2009}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{1821}{2009}\right)\) | \(e\left(\frac{1518}{2009}\right)\) | \(e\left(\frac{1997}{2009}\right)\) | \(e\left(\frac{1307}{2009}\right)\) | \(e\left(\frac{1727}{2009}\right)\) | \(e\left(\frac{4}{49}\right)\) | \(e\left(\frac{1424}{2009}\right)\) | \(e\left(\frac{1583}{2009}\right)\) |
| \(\chi_{4019}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{2009}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{4}{2009}\right)\) | \(e\left(\frac{1592}{2009}\right)\) | \(e\left(\frac{43}{2009}\right)\) | \(e\left(\frac{1511}{2009}\right)\) | \(e\left(\frac{6}{2009}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{1594}{2009}\right)\) | \(e\left(\frac{522}{2009}\right)\) |
| \(\chi_{4019}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{192}{2009}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{384}{2009}\right)\) | \(e\left(\frac{148}{2009}\right)\) | \(e\left(\frac{110}{2009}\right)\) | \(e\left(\frac{408}{2009}\right)\) | \(e\left(\frac{576}{2009}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{340}{2009}\right)\) | \(e\left(\frac{1896}{2009}\right)\) |
| \(\chi_{4019}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{2009}\right)\) | \(e\left(\frac{41}{49}\right)\) | \(e\left(\frac{262}{2009}\right)\) | \(e\left(\frac{1817}{2009}\right)\) | \(e\left(\frac{1812}{2009}\right)\) | \(e\left(\frac{1534}{2009}\right)\) | \(e\left(\frac{393}{2009}\right)\) | \(e\left(\frac{33}{49}\right)\) | \(e\left(\frac{1948}{2009}\right)\) | \(e\left(\frac{38}{2009}\right)\) |
| \(\chi_{4019}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{846}{2009}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{1692}{2009}\right)\) | \(e\left(\frac{401}{2009}\right)\) | \(e\left(\frac{108}{2009}\right)\) | \(e\left(\frac{291}{2009}\right)\) | \(e\left(\frac{529}{2009}\right)\) | \(e\left(\frac{13}{49}\right)\) | \(e\left(\frac{1247}{2009}\right)\) | \(e\left(\frac{1825}{2009}\right)\) |
| \(\chi_{4019}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{796}{2009}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{1592}{2009}\right)\) | \(e\left(\frac{781}{2009}\right)\) | \(e\left(\frac{1042}{2009}\right)\) | \(e\left(\frac{687}{2009}\right)\) | \(e\left(\frac{379}{2009}\right)\) | \(e\left(\frac{12}{49}\right)\) | \(e\left(\frac{1577}{2009}\right)\) | \(e\left(\frac{829}{2009}\right)\) |
| \(\chi_{4019}(26,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{904}{2009}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{1808}{2009}\right)\) | \(e\left(\frac{362}{2009}\right)\) | \(e\left(\frac{1355}{2009}\right)\) | \(e\left(\frac{1921}{2009}\right)\) | \(e\left(\frac{703}{2009}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{1266}{2009}\right)\) | \(e\left(\frac{891}{2009}\right)\) |
| \(\chi_{4019}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1026}{2009}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{43}{2009}\right)\) | \(e\left(\frac{1042}{2009}\right)\) | \(e\left(\frac{1969}{2009}\right)\) | \(e\left(\frac{1678}{2009}\right)\) | \(e\left(\frac{1069}{2009}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{59}{2009}\right)\) | \(e\left(\frac{589}{2009}\right)\) |
| \(\chi_{4019}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1058}{2009}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{107}{2009}\right)\) | \(e\left(\frac{397}{2009}\right)\) | \(e\left(\frac{648}{2009}\right)\) | \(e\left(\frac{1746}{2009}\right)\) | \(e\left(\frac{1165}{2009}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{1455}{2009}\right)\) | \(e\left(\frac{905}{2009}\right)\) |
| \(\chi_{4019}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1423}{2009}\right)\) | \(e\left(\frac{1}{49}\right)\) | \(e\left(\frac{837}{2009}\right)\) | \(e\left(\frac{1641}{2009}\right)\) | \(e\left(\frac{1464}{2009}\right)\) | \(e\left(\frac{1266}{2009}\right)\) | \(e\left(\frac{251}{2009}\right)\) | \(e\left(\frac{2}{49}\right)\) | \(e\left(\frac{1055}{2009}\right)\) | \(e\left(\frac{1747}{2009}\right)\) |
| \(\chi_{4019}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1760}{2009}\right)\) | \(e\left(\frac{47}{49}\right)\) | \(e\left(\frac{1511}{2009}\right)\) | \(e\left(\frac{687}{2009}\right)\) | \(e\left(\frac{1678}{2009}\right)\) | \(e\left(\frac{1731}{2009}\right)\) | \(e\left(\frac{1262}{2009}\right)\) | \(e\left(\frac{45}{49}\right)\) | \(e\left(\frac{438}{2009}\right)\) | \(e\left(\frac{1308}{2009}\right)\) |
| \(\chi_{4019}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{832}{2009}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{1664}{2009}\right)\) | \(e\left(\frac{1311}{2009}\right)\) | \(e\left(\frac{1816}{2009}\right)\) | \(e\left(\frac{1768}{2009}\right)\) | \(e\left(\frac{487}{2009}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{134}{2009}\right)\) | \(e\left(\frac{180}{2009}\right)\) |
| \(\chi_{4019}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1709}{2009}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{1409}{2009}\right)\) | \(e\left(\frac{271}{2009}\right)\) | \(e\left(\frac{1586}{2009}\right)\) | \(e\left(\frac{367}{2009}\right)\) | \(e\left(\frac{1109}{2009}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{1980}{2009}\right)\) | \(e\left(\frac{51}{2009}\right)\) |
| \(\chi_{4019}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{979}{2009}\right)\) | \(e\left(\frac{24}{49}\right)\) | \(e\left(\frac{1958}{2009}\right)\) | \(e\left(\frac{1801}{2009}\right)\) | \(e\left(\frac{1963}{2009}\right)\) | \(e\left(\frac{1327}{2009}\right)\) | \(e\left(\frac{928}{2009}\right)\) | \(e\left(\frac{48}{49}\right)\) | \(e\left(\frac{771}{2009}\right)\) | \(e\left(\frac{376}{2009}\right)\) |
| \(\chi_{4019}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1916}{2009}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{1823}{2009}\right)\) | \(e\left(\frac{305}{2009}\right)\) | \(e\left(\frac{1014}{2009}\right)\) | \(e\left(\frac{1058}{2009}\right)\) | \(e\left(\frac{1730}{2009}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{212}{2009}\right)\) | \(e\left(\frac{1844}{2009}\right)\) |
| \(\chi_{4019}(62,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{838}{2009}\right)\) | \(e\left(\frac{27}{49}\right)\) | \(e\left(\frac{1676}{2009}\right)\) | \(e\left(\frac{60}{2009}\right)\) | \(e\left(\frac{1945}{2009}\right)\) | \(e\left(\frac{274}{2009}\right)\) | \(e\left(\frac{505}{2009}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{898}{2009}\right)\) | \(e\left(\frac{1746}{2009}\right)\) |
| \(\chi_{4019}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{2009}\right)\) | \(e\left(\frac{26}{49}\right)\) | \(e\left(\frac{6}{2009}\right)\) | \(e\left(\frac{379}{2009}\right)\) | \(e\left(\frac{1069}{2009}\right)\) | \(e\left(\frac{1262}{2009}\right)\) | \(e\left(\frac{9}{2009}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{382}{2009}\right)\) | \(e\left(\frac{783}{2009}\right)\) |
| \(\chi_{4019}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1648}{2009}\right)\) | \(e\left(\frac{40}{49}\right)\) | \(e\left(\frac{1287}{2009}\right)\) | \(e\left(\frac{1940}{2009}\right)\) | \(e\left(\frac{1279}{2009}\right)\) | \(e\left(\frac{1493}{2009}\right)\) | \(e\left(\frac{926}{2009}\right)\) | \(e\left(\frac{31}{49}\right)\) | \(e\left(\frac{1579}{2009}\right)\) | \(e\left(\frac{202}{2009}\right)\) |
| \(\chi_{4019}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{2009}\right)\) | \(e\left(\frac{22}{49}\right)\) | \(e\left(\frac{186}{2009}\right)\) | \(e\left(\frac{1704}{2009}\right)\) | \(e\left(\frac{995}{2009}\right)\) | \(e\left(\frac{951}{2009}\right)\) | \(e\left(\frac{279}{2009}\right)\) | \(e\left(\frac{44}{49}\right)\) | \(e\left(\frac{1797}{2009}\right)\) | \(e\left(\frac{165}{2009}\right)\) |
| \(\chi_{4019}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{354}{2009}\right)\) | \(e\left(\frac{30}{49}\right)\) | \(e\left(\frac{708}{2009}\right)\) | \(e\left(\frac{524}{2009}\right)\) | \(e\left(\frac{1584}{2009}\right)\) | \(e\left(\frac{250}{2009}\right)\) | \(e\left(\frac{1062}{2009}\right)\) | \(e\left(\frac{11}{49}\right)\) | \(e\left(\frac{878}{2009}\right)\) | \(e\left(\frac{1989}{2009}\right)\) |
| \(\chi_{4019}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{274}{2009}\right)\) | \(e\left(\frac{39}{49}\right)\) | \(e\left(\frac{548}{2009}\right)\) | \(e\left(\frac{1132}{2009}\right)\) | \(e\left(\frac{1873}{2009}\right)\) | \(e\left(\frac{80}{2009}\right)\) | \(e\left(\frac{822}{2009}\right)\) | \(e\left(\frac{29}{49}\right)\) | \(e\left(\frac{1406}{2009}\right)\) | \(e\left(\frac{1199}{2009}\right)\) |
| \(\chi_{4019}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{477}{2009}\right)\) | \(e\left(\frac{18}{49}\right)\) | \(e\left(\frac{954}{2009}\right)\) | \(e\left(\frac{2000}{2009}\right)\) | \(e\left(\frac{1215}{2009}\right)\) | \(e\left(\frac{1767}{2009}\right)\) | \(e\left(\frac{1431}{2009}\right)\) | \(e\left(\frac{36}{49}\right)\) | \(e\left(\frac{468}{2009}\right)\) | \(e\left(\frac{1948}{2009}\right)\) |
| \(\chi_{4019}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1317}{2009}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{625}{2009}\right)\) | \(e\left(\frac{1643}{2009}\right)\) | \(e\left(\frac{1194}{2009}\right)\) | \(e\left(\frac{1543}{2009}\right)\) | \(e\left(\frac{1942}{2009}\right)\) | \(e\left(\frac{43}{49}\right)\) | \(e\left(\frac{951}{2009}\right)\) | \(e\left(\frac{198}{2009}\right)\) |
| \(\chi_{4019}(75,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{304}{2009}\right)\) | \(e\left(\frac{5}{49}\right)\) | \(e\left(\frac{608}{2009}\right)\) | \(e\left(\frac{904}{2009}\right)\) | \(e\left(\frac{509}{2009}\right)\) | \(e\left(\frac{646}{2009}\right)\) | \(e\left(\frac{912}{2009}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{1208}{2009}\right)\) | \(e\left(\frac{993}{2009}\right)\) |
| \(\chi_{4019}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{2009}\right)\) | \(e\left(\frac{23}{49}\right)\) | \(e\left(\frac{386}{2009}\right)\) | \(e\left(\frac{944}{2009}\right)\) | \(e\left(\frac{1136}{2009}\right)\) | \(e\left(\frac{159}{2009}\right)\) | \(e\left(\frac{579}{2009}\right)\) | \(e\left(\frac{46}{49}\right)\) | \(e\left(\frac{1137}{2009}\right)\) | \(e\left(\frac{148}{2009}\right)\) |
| \(\chi_{4019}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{2009}\right)\) | \(e\left(\frac{3}{49}\right)\) | \(e\left(\frac{12}{2009}\right)\) | \(e\left(\frac{758}{2009}\right)\) | \(e\left(\frac{129}{2009}\right)\) | \(e\left(\frac{515}{2009}\right)\) | \(e\left(\frac{18}{2009}\right)\) | \(e\left(\frac{6}{49}\right)\) | \(e\left(\frac{764}{2009}\right)\) | \(e\left(\frac{1566}{2009}\right)\) |
| \(\chi_{4019}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{412}{2009}\right)\) | \(e\left(\frac{10}{49}\right)\) | \(e\left(\frac{824}{2009}\right)\) | \(e\left(\frac{485}{2009}\right)\) | \(e\left(\frac{822}{2009}\right)\) | \(e\left(\frac{1880}{2009}\right)\) | \(e\left(\frac{1236}{2009}\right)\) | \(e\left(\frac{20}{49}\right)\) | \(e\left(\frac{897}{2009}\right)\) | \(e\left(\frac{1055}{2009}\right)\) |