from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
M = H._module
chi = DirichletCharacter(H, M([21175,9075,9723]))
chi.galois_orbit()
[g,chi] = znchar(Mod(41,586971))
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: | \(25410\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{12705})$ |
| Fixed field: | Number field defined by a degree 25410 polynomial (not computed) |
First 13 of 5280 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(13\) | \(16\) | \(17\) | \(19\) | \(20\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{586971}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6374}{12705}\right)\) | \(e\left(\frac{43}{12705}\right)\) | \(e\left(\frac{11821}{12705}\right)\) | \(e\left(\frac{2139}{4235}\right)\) | \(e\left(\frac{366}{847}\right)\) | \(e\left(\frac{109}{12705}\right)\) | \(e\left(\frac{86}{12705}\right)\) | \(e\left(\frac{2279}{8470}\right)\) | \(e\left(\frac{47}{605}\right)\) | \(e\left(\frac{11864}{12705}\right)\) |
| \(\chi_{586971}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9682}{12705}\right)\) | \(e\left(\frac{6659}{12705}\right)\) | \(e\left(\frac{5813}{12705}\right)\) | \(e\left(\frac{1212}{4235}\right)\) | \(e\left(\frac{186}{847}\right)\) | \(e\left(\frac{2402}{12705}\right)\) | \(e\left(\frac{613}{12705}\right)\) | \(e\left(\frac{5657}{8470}\right)\) | \(e\left(\frac{131}{605}\right)\) | \(e\left(\frac{12472}{12705}\right)\) |
| \(\chi_{586971}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1598}{12705}\right)\) | \(e\left(\frac{3196}{12705}\right)\) | \(e\left(\frac{7867}{12705}\right)\) | \(e\left(\frac{1598}{4235}\right)\) | \(e\left(\frac{631}{847}\right)\) | \(e\left(\frac{12238}{12705}\right)\) | \(e\left(\frac{6392}{12705}\right)\) | \(e\left(\frac{4223}{8470}\right)\) | \(e\left(\frac{4}{605}\right)\) | \(e\left(\frac{11063}{12705}\right)\) |
| \(\chi_{586971}(272,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2014}{12705}\right)\) | \(e\left(\frac{4028}{12705}\right)\) | \(e\left(\frac{8786}{12705}\right)\) | \(e\left(\frac{2014}{4235}\right)\) | \(e\left(\frac{720}{847}\right)\) | \(e\left(\frac{6074}{12705}\right)\) | \(e\left(\frac{8056}{12705}\right)\) | \(e\left(\frac{5969}{8470}\right)\) | \(e\left(\frac{27}{605}\right)\) | \(e\left(\frac{109}{12705}\right)\) |
| \(\chi_{586971}(398,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2803}{12705}\right)\) | \(e\left(\frac{5606}{12705}\right)\) | \(e\left(\frac{12392}{12705}\right)\) | \(e\left(\frac{2803}{4235}\right)\) | \(e\left(\frac{166}{847}\right)\) | \(e\left(\frac{6233}{12705}\right)\) | \(e\left(\frac{11212}{12705}\right)\) | \(e\left(\frac{4903}{8470}\right)\) | \(e\left(\frac{584}{605}\right)\) | \(e\left(\frac{5293}{12705}\right)\) |
| \(\chi_{586971}(545,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11567}{12705}\right)\) | \(e\left(\frac{10429}{12705}\right)\) | \(e\left(\frac{7198}{12705}\right)\) | \(e\left(\frac{3097}{4235}\right)\) | \(e\left(\frac{404}{847}\right)\) | \(e\left(\frac{7822}{12705}\right)\) | \(e\left(\frac{8153}{12705}\right)\) | \(e\left(\frac{2187}{8470}\right)\) | \(e\left(\frac{481}{605}\right)\) | \(e\left(\frac{4922}{12705}\right)\) |
| \(\chi_{586971}(776,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7312}{12705}\right)\) | \(e\left(\frac{1919}{12705}\right)\) | \(e\left(\frac{10778}{12705}\right)\) | \(e\left(\frac{3077}{4235}\right)\) | \(e\left(\frac{359}{847}\right)\) | \(e\left(\frac{137}{12705}\right)\) | \(e\left(\frac{3838}{12705}\right)\) | \(e\left(\frac{67}{8470}\right)\) | \(e\left(\frac{381}{605}\right)\) | \(e\left(\frac{12697}{12705}\right)\) |
| \(\chi_{586971}(860,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{593}{12705}\right)\) | \(e\left(\frac{1186}{12705}\right)\) | \(e\left(\frac{9892}{12705}\right)\) | \(e\left(\frac{593}{4235}\right)\) | \(e\left(\frac{699}{847}\right)\) | \(e\left(\frac{10393}{12705}\right)\) | \(e\left(\frac{2372}{12705}\right)\) | \(e\left(\frac{7803}{8470}\right)\) | \(e\left(\frac{424}{605}\right)\) | \(e\left(\frac{11078}{12705}\right)\) |
| \(\chi_{586971}(986,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12281}{12705}\right)\) | \(e\left(\frac{11857}{12705}\right)\) | \(e\left(\frac{11524}{12705}\right)\) | \(e\left(\frac{3811}{4235}\right)\) | \(e\left(\frac{740}{847}\right)\) | \(e\left(\frac{1396}{12705}\right)\) | \(e\left(\frac{11009}{12705}\right)\) | \(e\left(\frac{1641}{8470}\right)\) | \(e\left(\frac{58}{605}\right)\) | \(e\left(\frac{10676}{12705}\right)\) |
| \(\chi_{586971}(1091,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9883}{12705}\right)\) | \(e\left(\frac{7061}{12705}\right)\) | \(e\left(\frac{2867}{12705}\right)\) | \(e\left(\frac{1413}{4235}\right)\) | \(e\left(\frac{3}{847}\right)\) | \(e\left(\frac{7853}{12705}\right)\) | \(e\left(\frac{1417}{12705}\right)\) | \(e\left(\frac{1553}{8470}\right)\) | \(e\left(\frac{289}{605}\right)\) | \(e\left(\frac{9928}{12705}\right)\) |
| \(\chi_{586971}(1217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4666}{12705}\right)\) | \(e\left(\frac{9332}{12705}\right)\) | \(e\left(\frac{6704}{12705}\right)\) | \(e\left(\frac{431}{4235}\right)\) | \(e\left(\frac{758}{847}\right)\) | \(e\left(\frac{11246}{12705}\right)\) | \(e\left(\frac{5959}{12705}\right)\) | \(e\left(\frac{7571}{8470}\right)\) | \(e\left(\frac{98}{605}\right)\) | \(e\left(\frac{3331}{12705}\right)\) |
| \(\chi_{586971}(1238,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1112}{12705}\right)\) | \(e\left(\frac{2224}{12705}\right)\) | \(e\left(\frac{11008}{12705}\right)\) | \(e\left(\frac{1112}{4235}\right)\) | \(e\left(\frac{808}{847}\right)\) | \(e\left(\frac{2092}{12705}\right)\) | \(e\left(\frac{4448}{12705}\right)\) | \(e\left(\frac{3527}{8470}\right)\) | \(e\left(\frac{236}{605}\right)\) | \(e\left(\frac{527}{12705}\right)\) |
| \(\chi_{586971}(1427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1004}{12705}\right)\) | \(e\left(\frac{2008}{12705}\right)\) | \(e\left(\frac{7471}{12705}\right)\) | \(e\left(\frac{1004}{4235}\right)\) | \(e\left(\frac{565}{847}\right)\) | \(e\left(\frac{1249}{12705}\right)\) | \(e\left(\frac{4016}{12705}\right)\) | \(e\left(\frac{549}{8470}\right)\) | \(e\left(\frac{422}{605}\right)\) | \(e\left(\frac{9479}{12705}\right)\) |