from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
M = H._module
chi = DirichletCharacter(H, M([16940,9680,7728]))
chi.galois_orbit()
[g,chi] = znchar(Mod(25,586971))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(586971\) | |
| Conductor: | \(586971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12705\) | 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: | $\Q(\zeta_{12705})$ |
| Fixed field: | Number field defined by a degree 12705 polynomial (not computed) |
First 17 of 5280 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{586971}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3708}{4235}\right)\) | \(e\left(\frac{3181}{4235}\right)\) | \(e\left(\frac{4336}{12705}\right)\) | \(e\left(\frac{2654}{4235}\right)\) | \(e\left(\frac{551}{2541}\right)\) | \(e\left(\frac{2129}{12705}\right)\) | \(e\left(\frac{2127}{4235}\right)\) | \(e\left(\frac{11191}{12705}\right)\) | \(e\left(\frac{1211}{1815}\right)\) | \(e\left(\frac{1174}{12705}\right)\) |
| \(\chi_{586971}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3638}{4235}\right)\) | \(e\left(\frac{3041}{4235}\right)\) | \(e\left(\frac{12281}{12705}\right)\) | \(e\left(\frac{2444}{4235}\right)\) | \(e\left(\frac{2098}{2541}\right)\) | \(e\left(\frac{8254}{12705}\right)\) | \(e\left(\frac{1847}{4235}\right)\) | \(e\left(\frac{11681}{12705}\right)\) | \(e\left(\frac{481}{1815}\right)\) | \(e\left(\frac{8699}{12705}\right)\) |
| \(\chi_{586971}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1304}{4235}\right)\) | \(e\left(\frac{2608}{4235}\right)\) | \(e\left(\frac{5303}{12705}\right)\) | \(e\left(\frac{3912}{4235}\right)\) | \(e\left(\frac{1843}{2541}\right)\) | \(e\left(\frac{2302}{12705}\right)\) | \(e\left(\frac{981}{4235}\right)\) | \(e\left(\frac{3698}{12705}\right)\) | \(e\left(\frac{358}{1815}\right)\) | \(e\left(\frac{422}{12705}\right)\) |
| \(\chi_{586971}(466,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4002}{4235}\right)\) | \(e\left(\frac{3769}{4235}\right)\) | \(e\left(\frac{1459}{12705}\right)\) | \(e\left(\frac{3536}{4235}\right)\) | \(e\left(\frac{152}{2541}\right)\) | \(e\left(\frac{6896}{12705}\right)\) | \(e\left(\frac{3303}{4235}\right)\) | \(e\left(\frac{11674}{12705}\right)\) | \(e\left(\frac{284}{1815}\right)\) | \(e\left(\frac{61}{12705}\right)\) |
| \(\chi_{586971}(499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{292}{4235}\right)\) | \(e\left(\frac{584}{4235}\right)\) | \(e\left(\frac{7514}{12705}\right)\) | \(e\left(\frac{876}{4235}\right)\) | \(e\left(\frac{1678}{2541}\right)\) | \(e\left(\frac{9661}{12705}\right)\) | \(e\left(\frac{1168}{4235}\right)\) | \(e\left(\frac{3764}{12705}\right)\) | \(e\left(\frac{919}{1815}\right)\) | \(e\left(\frac{9266}{12705}\right)\) |
| \(\chi_{586971}(592,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3406}{4235}\right)\) | \(e\left(\frac{2577}{4235}\right)\) | \(e\left(\frac{6427}{12705}\right)\) | \(e\left(\frac{1748}{4235}\right)\) | \(e\left(\frac{788}{2541}\right)\) | \(e\left(\frac{8468}{12705}\right)\) | \(e\left(\frac{919}{4235}\right)\) | \(e\left(\frac{6892}{12705}\right)\) | \(e\left(\frac{1052}{1815}\right)\) | \(e\left(\frac{1453}{12705}\right)\) |
| \(\chi_{586971}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3181}{4235}\right)\) | \(e\left(\frac{2127}{4235}\right)\) | \(e\left(\frac{8672}{12705}\right)\) | \(e\left(\frac{1073}{4235}\right)\) | \(e\left(\frac{1102}{2541}\right)\) | \(e\left(\frac{4258}{12705}\right)\) | \(e\left(\frac{19}{4235}\right)\) | \(e\left(\frac{9677}{12705}\right)\) | \(e\left(\frac{607}{1815}\right)\) | \(e\left(\frac{2348}{12705}\right)\) |
| \(\chi_{586971}(718,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1193}{4235}\right)\) | \(e\left(\frac{2386}{4235}\right)\) | \(e\left(\frac{8161}{12705}\right)\) | \(e\left(\frac{3579}{4235}\right)\) | \(e\left(\frac{2348}{2541}\right)\) | \(e\left(\frac{2879}{12705}\right)\) | \(e\left(\frac{537}{4235}\right)\) | \(e\left(\frac{5806}{12705}\right)\) | \(e\left(\frac{566}{1815}\right)\) | \(e\left(\frac{2614}{12705}\right)\) |
| \(\chi_{586971}(751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2773}{4235}\right)\) | \(e\left(\frac{1311}{4235}\right)\) | \(e\left(\frac{6701}{12705}\right)\) | \(e\left(\frac{4084}{4235}\right)\) | \(e\left(\frac{463}{2541}\right)\) | \(e\left(\frac{11644}{12705}\right)\) | \(e\left(\frac{2622}{4235}\right)\) | \(e\left(\frac{5636}{12705}\right)\) | \(e\left(\frac{1486}{1815}\right)\) | \(e\left(\frac{10634}{12705}\right)\) |
| \(\chi_{586971}(907,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3379}{4235}\right)\) | \(e\left(\frac{2523}{4235}\right)\) | \(e\left(\frac{598}{12705}\right)\) | \(e\left(\frac{1667}{4235}\right)\) | \(e\left(\frac{2147}{2541}\right)\) | \(e\left(\frac{11012}{12705}\right)\) | \(e\left(\frac{811}{4235}\right)\) | \(e\left(\frac{6718}{12705}\right)\) | \(e\left(\frac{563}{1815}\right)\) | \(e\left(\frac{8167}{12705}\right)\) |
| \(\chi_{586971}(940,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1874}{4235}\right)\) | \(e\left(\frac{3748}{4235}\right)\) | \(e\left(\frac{4133}{12705}\right)\) | \(e\left(\frac{1387}{4235}\right)\) | \(e\left(\frac{1951}{2541}\right)\) | \(e\left(\frac{5062}{12705}\right)\) | \(e\left(\frac{3261}{4235}\right)\) | \(e\left(\frac{8783}{12705}\right)\) | \(e\left(\frac{598}{1815}\right)\) | \(e\left(\frac{2672}{12705}\right)\) |
| \(\chi_{586971}(1159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3587}{4235}\right)\) | \(e\left(\frac{2939}{4235}\right)\) | \(e\left(\frac{8329}{12705}\right)\) | \(e\left(\frac{2291}{4235}\right)\) | \(e\left(\frac{1277}{2541}\right)\) | \(e\left(\frac{1766}{12705}\right)\) | \(e\left(\frac{1643}{4235}\right)\) | \(e\left(\frac{4294}{12705}\right)\) | \(e\left(\frac{1574}{1815}\right)\) | \(e\left(\frac{4441}{12705}\right)\) |
| \(\chi_{586971}(1192,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3452}{4235}\right)\) | \(e\left(\frac{2669}{4235}\right)\) | \(e\left(\frac{359}{12705}\right)\) | \(e\left(\frac{1886}{4235}\right)\) | \(e\left(\frac{2143}{2541}\right)\) | \(e\left(\frac{6016}{12705}\right)\) | \(e\left(\frac{1103}{4235}\right)\) | \(e\left(\frac{11894}{12705}\right)\) | \(e\left(\frac{1549}{1815}\right)\) | \(e\left(\frac{8366}{12705}\right)\) |
| \(\chi_{586971}(1285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{716}{4235}\right)\) | \(e\left(\frac{1432}{4235}\right)\) | \(e\left(\frac{2587}{12705}\right)\) | \(e\left(\frac{2148}{4235}\right)\) | \(e\left(\frac{947}{2541}\right)\) | \(e\left(\frac{1238}{12705}\right)\) | \(e\left(\frac{2864}{4235}\right)\) | \(e\left(\frac{6967}{12705}\right)\) | \(e\left(\frac{1607}{1815}\right)\) | \(e\left(\frac{6883}{12705}\right)\) |
| \(\chi_{586971}(1318,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3296}{4235}\right)\) | \(e\left(\frac{2357}{4235}\right)\) | \(e\left(\frac{10442}{12705}\right)\) | \(e\left(\frac{1418}{4235}\right)\) | \(e\left(\frac{1525}{2541}\right)\) | \(e\left(\frac{6598}{12705}\right)\) | \(e\left(\frac{479}{4235}\right)\) | \(e\left(\frac{1007}{12705}\right)\) | \(e\left(\frac{337}{1815}\right)\) | \(e\left(\frac{4808}{12705}\right)\) |
| \(\chi_{586971}(1411,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3683}{4235}\right)\) | \(e\left(\frac{3131}{4235}\right)\) | \(e\left(\frac{5056}{12705}\right)\) | \(e\left(\frac{2579}{4235}\right)\) | \(e\left(\frac{680}{2541}\right)\) | \(e\left(\frac{8249}{12705}\right)\) | \(e\left(\frac{2027}{4235}\right)\) | \(e\left(\frac{11971}{12705}\right)\) | \(e\left(\frac{86}{1815}\right)\) | \(e\left(\frac{1744}{12705}\right)\) |
| \(\chi_{586971}(1444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2678}{4235}\right)\) | \(e\left(\frac{1121}{4235}\right)\) | \(e\left(\frac{6896}{12705}\right)\) | \(e\left(\frac{3799}{4235}\right)\) | \(e\left(\frac{445}{2541}\right)\) | \(e\left(\frac{6949}{12705}\right)\) | \(e\left(\frac{2242}{4235}\right)\) | \(e\left(\frac{11141}{12705}\right)\) | \(e\left(\frac{841}{1815}\right)\) | \(e\left(\frac{10259}{12705}\right)\) |