from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6724, base_ring=CyclotomicField(1640))
M = H._module
chi = DirichletCharacter(H, M([0,1511]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,6724))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6724\) | |
| Conductor: | \(1681\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1640\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1681.p | 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: | $\Q(\zeta_{1640})$ |
| Fixed field: | Number field defined by a degree 1640 polynomial (not computed) |
First 31 of 640 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6724}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{213}{328}\right)\) | \(e\left(\frac{601}{820}\right)\) | \(e\left(\frac{49}{1640}\right)\) | \(e\left(\frac{49}{164}\right)\) | \(e\left(\frac{693}{1640}\right)\) | \(e\left(\frac{241}{1640}\right)\) | \(e\left(\frac{627}{1640}\right)\) | \(e\left(\frac{903}{1640}\right)\) | \(e\left(\frac{919}{1640}\right)\) | \(e\left(\frac{557}{820}\right)\) |
| \(\chi_{6724}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{328}\right)\) | \(e\left(\frac{503}{820}\right)\) | \(e\left(\frac{1007}{1640}\right)\) | \(e\left(\frac{23}{164}\right)\) | \(e\left(\frac{419}{1640}\right)\) | \(e\left(\frac{903}{1640}\right)\) | \(e\left(\frac{301}{1640}\right)\) | \(e\left(\frac{49}{1640}\right)\) | \(e\left(\frac{177}{1640}\right)\) | \(e\left(\frac{151}{820}\right)\) |
| \(\chi_{6724}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{328}\right)\) | \(e\left(\frac{417}{820}\right)\) | \(e\left(\frac{1513}{1640}\right)\) | \(e\left(\frac{37}{164}\right)\) | \(e\left(\frac{781}{1640}\right)\) | \(e\left(\frac{1417}{1640}\right)\) | \(e\left(\frac{1019}{1640}\right)\) | \(e\left(\frac{471}{1640}\right)\) | \(e\left(\frac{463}{1640}\right)\) | \(e\left(\frac{29}{820}\right)\) |
| \(\chi_{6724}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{328}\right)\) | \(e\left(\frac{817}{820}\right)\) | \(e\left(\frac{933}{1640}\right)\) | \(e\left(\frac{113}{164}\right)\) | \(e\left(\frac{1481}{1640}\right)\) | \(e\left(\frac{37}{1640}\right)\) | \(e\left(\frac{559}{1640}\right)\) | \(e\left(\frac{91}{1640}\right)\) | \(e\left(\frac{563}{1640}\right)\) | \(e\left(\frac{749}{820}\right)\) |
| \(\chi_{6724}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{328}\right)\) | \(e\left(\frac{703}{820}\right)\) | \(e\left(\frac{307}{1640}\right)\) | \(e\left(\frac{143}{164}\right)\) | \(e\left(\frac{359}{1640}\right)\) | \(e\left(\frac{1443}{1640}\right)\) | \(e\left(\frac{481}{1640}\right)\) | \(e\left(\frac{269}{1640}\right)\) | \(e\left(\frac{637}{1640}\right)\) | \(e\left(\frac{511}{820}\right)\) |
| \(\chi_{6724}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{328}\right)\) | \(e\left(\frac{421}{820}\right)\) | \(e\left(\frac{269}{1640}\right)\) | \(e\left(\frac{105}{164}\right)\) | \(e\left(\frac{993}{1640}\right)\) | \(e\left(\frac{821}{1640}\right)\) | \(e\left(\frac{1367}{1640}\right)\) | \(e\left(\frac{1443}{1640}\right)\) | \(e\left(\frac{259}{1640}\right)\) | \(e\left(\frac{397}{820}\right)\) |
| \(\chi_{6724}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{285}{328}\right)\) | \(e\left(\frac{229}{820}\right)\) | \(e\left(\frac{121}{1640}\right)\) | \(e\left(\frac{121}{164}\right)\) | \(e\left(\frac{1477}{1640}\right)\) | \(e\left(\frac{729}{1640}\right)\) | \(e\left(\frac{243}{1640}\right)\) | \(e\left(\frac{1527}{1640}\right)\) | \(e\left(\frac{1031}{1640}\right)\) | \(e\left(\frac{773}{820}\right)\) |
| \(\chi_{6724}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{328}\right)\) | \(e\left(\frac{493}{820}\right)\) | \(e\left(\frac{837}{1640}\right)\) | \(e\left(\frac{17}{164}\right)\) | \(e\left(\frac{1529}{1640}\right)\) | \(e\left(\frac{1573}{1640}\right)\) | \(e\left(\frac{1071}{1640}\right)\) | \(e\left(\frac{899}{1640}\right)\) | \(e\left(\frac{1507}{1640}\right)\) | \(e\left(\frac{461}{820}\right)\) |
| \(\chi_{6724}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{328}\right)\) | \(e\left(\frac{247}{820}\right)\) | \(e\left(\frac{1083}{1640}\right)\) | \(e\left(\frac{99}{164}\right)\) | \(e\left(\frac{791}{1640}\right)\) | \(e\left(\frac{507}{1640}\right)\) | \(e\left(\frac{169}{1640}\right)\) | \(e\left(\frac{981}{1640}\right)\) | \(e\left(\frac{933}{1640}\right)\) | \(e\left(\frac{379}{820}\right)\) |
| \(\chi_{6724}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{328}\right)\) | \(e\left(\frac{759}{820}\right)\) | \(e\left(\frac{111}{1640}\right)\) | \(e\left(\frac{111}{164}\right)\) | \(e\left(\frac{867}{1640}\right)\) | \(e\left(\frac{479}{1640}\right)\) | \(e\left(\frac{1253}{1640}\right)\) | \(e\left(\frac{1577}{1640}\right)\) | \(e\left(\frac{241}{1640}\right)\) | \(e\left(\frac{743}{820}\right)\) |
| \(\chi_{6724}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{328}\right)\) | \(e\left(\frac{251}{820}\right)\) | \(e\left(\frac{659}{1640}\right)\) | \(e\left(\frac{3}{164}\right)\) | \(e\left(\frac{183}{1640}\right)\) | \(e\left(\frac{731}{1640}\right)\) | \(e\left(\frac{1337}{1640}\right)\) | \(e\left(\frac{1133}{1640}\right)\) | \(e\left(\frac{1549}{1640}\right)\) | \(e\left(\frac{747}{820}\right)\) |
| \(\chi_{6724}(129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{328}\right)\) | \(e\left(\frac{191}{820}\right)\) | \(e\left(\frac{1279}{1640}\right)\) | \(e\left(\frac{131}{164}\right)\) | \(e\left(\frac{283}{1640}\right)\) | \(e\left(\frac{1471}{1640}\right)\) | \(e\left(\frac{1037}{1640}\right)\) | \(e\left(\frac{1313}{1640}\right)\) | \(e\left(\frac{1329}{1640}\right)\) | \(e\left(\frac{147}{820}\right)\) |
| \(\chi_{6724}(145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{295}{328}\right)\) | \(e\left(\frac{519}{820}\right)\) | \(e\left(\frac{131}{1640}\right)\) | \(e\left(\frac{131}{164}\right)\) | \(e\left(\frac{447}{1640}\right)\) | \(e\left(\frac{979}{1640}\right)\) | \(e\left(\frac{873}{1640}\right)\) | \(e\left(\frac{1477}{1640}\right)\) | \(e\left(\frac{181}{1640}\right)\) | \(e\left(\frac{803}{820}\right)\) |
| \(\chi_{6724}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{328}\right)\) | \(e\left(\frac{107}{820}\right)\) | \(e\left(\frac{343}{1640}\right)\) | \(e\left(\frac{15}{164}\right)\) | \(e\left(\frac{1571}{1640}\right)\) | \(e\left(\frac{47}{1640}\right)\) | \(e\left(\frac{1109}{1640}\right)\) | \(e\left(\frac{1401}{1640}\right)\) | \(e\left(\frac{1513}{1640}\right)\) | \(e\left(\frac{619}{820}\right)\) |
| \(\chi_{6724}(153,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{328}\right)\) | \(e\left(\frac{153}{820}\right)\) | \(e\left(\frac{1617}{1640}\right)\) | \(e\left(\frac{141}{164}\right)\) | \(e\left(\frac{1549}{1640}\right)\) | \(e\left(\frac{1393}{1640}\right)\) | \(e\left(\frac{1011}{1640}\right)\) | \(e\left(\frac{279}{1640}\right)\) | \(e\left(\frac{807}{1640}\right)\) | \(e\left(\frac{341}{820}\right)\) |
| \(\chi_{6724}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{105}{328}\right)\) | \(e\left(\frac{749}{820}\right)\) | \(e\left(\frac{1581}{1640}\right)\) | \(e\left(\frac{105}{164}\right)\) | \(e\left(\frac{337}{1640}\right)\) | \(e\left(\frac{1149}{1640}\right)\) | \(e\left(\frac{383}{1640}\right)\) | \(e\left(\frac{787}{1640}\right)\) | \(e\left(\frac{1571}{1640}\right)\) | \(e\left(\frac{233}{820}\right)\) |
| \(\chi_{6724}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{328}\right)\) | \(e\left(\frac{141}{820}\right)\) | \(e\left(\frac{1249}{1640}\right)\) | \(e\left(\frac{101}{164}\right)\) | \(e\left(\frac{93}{1640}\right)\) | \(e\left(\frac{721}{1640}\right)\) | \(e\left(\frac{787}{1640}\right)\) | \(e\left(\frac{1463}{1640}\right)\) | \(e\left(\frac{599}{1640}\right)\) | \(e\left(\frac{57}{820}\right)\) |
| \(\chi_{6724}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{328}\right)\) | \(e\left(\frac{103}{820}\right)\) | \(e\left(\frac{767}{1640}\right)\) | \(e\left(\frac{111}{164}\right)\) | \(e\left(\frac{539}{1640}\right)\) | \(e\left(\frac{1463}{1640}\right)\) | \(e\left(\frac{1581}{1640}\right)\) | \(e\left(\frac{1249}{1640}\right)\) | \(e\left(\frac{897}{1640}\right)\) | \(e\left(\frac{251}{820}\right)\) |
| \(\chi_{6724}(193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{213}{328}\right)\) | \(e\left(\frac{437}{820}\right)\) | \(e\left(\frac{1033}{1640}\right)\) | \(e\left(\frac{49}{164}\right)\) | \(e\left(\frac{1021}{1640}\right)\) | \(e\left(\frac{897}{1640}\right)\) | \(e\left(\frac{299}{1640}\right)\) | \(e\left(\frac{1231}{1640}\right)\) | \(e\left(\frac{263}{1640}\right)\) | \(e\left(\frac{229}{820}\right)\) |
| \(\chi_{6724}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{328}\right)\) | \(e\left(\frac{797}{820}\right)\) | \(e\left(\frac{1413}{1640}\right)\) | \(e\left(\frac{101}{164}\right)\) | \(e\left(\frac{1241}{1640}\right)\) | \(e\left(\frac{557}{1640}\right)\) | \(e\left(\frac{1279}{1640}\right)\) | \(e\left(\frac{971}{1640}\right)\) | \(e\left(\frac{763}{1640}\right)\) | \(e\left(\frac{549}{820}\right)\) |
| \(\chi_{6724}(229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{328}\right)\) | \(e\left(\frac{283}{820}\right)\) | \(e\left(\frac{547}{1640}\right)\) | \(e\left(\frac{55}{164}\right)\) | \(e\left(\frac{239}{1640}\right)\) | \(e\left(\frac{883}{1640}\right)\) | \(e\left(\frac{841}{1640}\right)\) | \(e\left(\frac{709}{1640}\right)\) | \(e\left(\frac{1557}{1640}\right)\) | \(e\left(\frac{411}{820}\right)\) |
| \(\chi_{6724}(233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{328}\right)\) | \(e\left(\frac{61}{820}\right)\) | \(e\left(\frac{709}{1640}\right)\) | \(e\left(\frac{53}{164}\right)\) | \(e\left(\frac{1593}{1640}\right)\) | \(e\left(\frac{341}{1640}\right)\) | \(e\left(\frac{1207}{1640}\right)\) | \(e\left(\frac{883}{1640}\right)\) | \(e\left(\frac{579}{1640}\right)\) | \(e\left(\frac{77}{820}\right)\) |
| \(\chi_{6724}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{328}\right)\) | \(e\left(\frac{429}{820}\right)\) | \(e\left(\frac{241}{1640}\right)\) | \(e\left(\frac{77}{164}\right)\) | \(e\left(\frac{597}{1640}\right)\) | \(e\left(\frac{449}{1640}\right)\) | \(e\left(\frac{1243}{1640}\right)\) | \(e\left(\frac{927}{1640}\right)\) | \(e\left(\frac{671}{1640}\right)\) | \(e\left(\frac{313}{820}\right)\) |
| \(\chi_{6724}(257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{328}\right)\) | \(e\left(\frac{173}{820}\right)\) | \(e\left(\frac{317}{1640}\right)\) | \(e\left(\frac{153}{164}\right)\) | \(e\left(\frac{969}{1640}\right)\) | \(e\left(\frac{53}{1640}\right)\) | \(e\left(\frac{1111}{1640}\right)\) | \(e\left(\frac{219}{1640}\right)\) | \(e\left(\frac{1427}{1640}\right)\) | \(e\left(\frac{541}{820}\right)\) |
| \(\chi_{6724}(261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{319}{328}\right)\) | \(e\left(\frac{67}{820}\right)\) | \(e\left(\frac{483}{1640}\right)\) | \(e\left(\frac{155}{164}\right)\) | \(e\left(\frac{271}{1640}\right)\) | \(e\left(\frac{267}{1640}\right)\) | \(e\left(\frac{89}{1640}\right)\) | \(e\left(\frac{701}{1640}\right)\) | \(e\left(\frac{1093}{1640}\right)\) | \(e\left(\frac{219}{820}\right)\) |
| \(\chi_{6724}(265,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{328}\right)\) | \(e\left(\frac{99}{820}\right)\) | \(e\left(\frac{1191}{1640}\right)\) | \(e\left(\frac{43}{164}\right)\) | \(e\left(\frac{1147}{1640}\right)\) | \(e\left(\frac{1239}{1640}\right)\) | \(e\left(\frac{413}{1640}\right)\) | \(e\left(\frac{1097}{1640}\right)\) | \(e\left(\frac{281}{1640}\right)\) | \(e\left(\frac{703}{820}\right)\) |
| \(\chi_{6724}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{328}\right)\) | \(e\left(\frac{291}{820}\right)\) | \(e\left(\frac{1339}{1640}\right)\) | \(e\left(\frac{27}{164}\right)\) | \(e\left(\frac{663}{1640}\right)\) | \(e\left(\frac{1331}{1640}\right)\) | \(e\left(\frac{1537}{1640}\right)\) | \(e\left(\frac{1013}{1640}\right)\) | \(e\left(\frac{1149}{1640}\right)\) | \(e\left(\frac{327}{820}\right)\) |
| \(\chi_{6724}(293,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{328}\right)\) | \(e\left(\frac{151}{820}\right)\) | \(e\left(\frac{599}{1640}\right)\) | \(e\left(\frac{107}{164}\right)\) | \(e\left(\frac{1443}{1640}\right)\) | \(e\left(\frac{871}{1640}\right)\) | \(e\left(\frac{837}{1640}\right)\) | \(e\left(\frac{1433}{1640}\right)\) | \(e\left(\frac{89}{1640}\right)\) | \(e\left(\frac{567}{820}\right)\) |
| \(\chi_{6724}(309,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{328}\right)\) | \(e\left(\frac{359}{820}\right)\) | \(e\left(\frac{691}{1640}\right)\) | \(e\left(\frac{35}{164}\right)\) | \(e\left(\frac{167}{1640}\right)\) | \(e\left(\frac{219}{1640}\right)\) | \(e\left(\frac{73}{1640}\right)\) | \(e\left(\frac{317}{1640}\right)\) | \(e\left(\frac{141}{1640}\right)\) | \(e\left(\frac{23}{820}\right)\) |
| \(\chi_{6724}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{328}\right)\) | \(e\left(\frac{473}{820}\right)\) | \(e\left(\frac{497}{1640}\right)\) | \(e\left(\frac{5}{164}\right)\) | \(e\left(\frac{469}{1640}\right)\) | \(e\left(\frac{1273}{1640}\right)\) | \(e\left(\frac{971}{1640}\right)\) | \(e\left(\frac{959}{1640}\right)\) | \(e\left(\frac{887}{1640}\right)\) | \(e\left(\frac{261}{820}\right)\) |
| \(\chi_{6724}(321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{328}\right)\) | \(e\left(\frac{549}{820}\right)\) | \(e\left(\frac{1461}{1640}\right)\) | \(e\left(\frac{149}{164}\right)\) | \(e\left(\frac{1217}{1640}\right)\) | \(e\left(\frac{1429}{1640}\right)\) | \(e\left(\frac{1023}{1640}\right)\) | \(e\left(\frac{1387}{1640}\right)\) | \(e\left(\frac{291}{1640}\right)\) | \(e\left(\frac{693}{820}\right)\) |