from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(43904, base_ring=CyclotomicField(4704))
M = H._module
chi = DirichletCharacter(H, M([0,3675,1184]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,43904))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(43904\) | |
| Conductor: | \(43904\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4704\) | 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: | $\Q(\zeta_{4704})$ |
| Fixed field: | Number field defined by a degree 4704 polynomial (not computed) |
First 31 of 1344 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{43904}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2801}{4704}\right)\) | \(e\left(\frac{379}{4704}\right)\) | \(e\left(\frac{449}{2352}\right)\) | \(e\left(\frac{3575}{4704}\right)\) | \(e\left(\frac{711}{1568}\right)\) | \(e\left(\frac{265}{392}\right)\) | \(e\left(\frac{197}{1176}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{509}{2352}\right)\) | \(e\left(\frac{379}{2352}\right)\) |
| \(\chi_{43904}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2365}{4704}\right)\) | \(e\left(\frac{671}{4704}\right)\) | \(e\left(\frac{13}{2352}\right)\) | \(e\left(\frac{1675}{4704}\right)\) | \(e\left(\frac{59}{1568}\right)\) | \(e\left(\frac{253}{392}\right)\) | \(e\left(\frac{265}{1176}\right)\) | \(e\left(\frac{89}{96}\right)\) | \(e\left(\frac{1801}{2352}\right)\) | \(e\left(\frac{671}{2352}\right)\) |
| \(\chi_{43904}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2291}{4704}\right)\) | \(e\left(\frac{2641}{4704}\right)\) | \(e\left(\frac{2291}{2352}\right)\) | \(e\left(\frac{101}{4704}\right)\) | \(e\left(\frac{85}{1568}\right)\) | \(e\left(\frac{19}{392}\right)\) | \(e\left(\frac{23}{1176}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{1319}{2352}\right)\) | \(e\left(\frac{289}{2352}\right)\) |
| \(\chi_{43904}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1375}{4704}\right)\) | \(e\left(\frac{3125}{4704}\right)\) | \(e\left(\frac{1375}{2352}\right)\) | \(e\left(\frac{1849}{4704}\right)\) | \(e\left(\frac{873}{1568}\right)\) | \(e\left(\frac{375}{392}\right)\) | \(e\left(\frac{619}{1176}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{883}{2352}\right)\) | \(e\left(\frac{773}{2352}\right)\) |
| \(\chi_{43904}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{773}{4704}\right)\) | \(e\left(\frac{3895}{4704}\right)\) | \(e\left(\frac{773}{2352}\right)\) | \(e\left(\frac{995}{4704}\right)\) | \(e\left(\frac{915}{1568}\right)\) | \(e\left(\frac{389}{392}\right)\) | \(e\left(\frac{17}{1176}\right)\) | \(e\left(\frac{1}{96}\right)\) | \(e\left(\frac{1793}{2352}\right)\) | \(e\left(\frac{1543}{2352}\right)\) |
| \(\chi_{43904}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2951}{4704}\right)\) | \(e\left(\frac{4141}{4704}\right)\) | \(e\left(\frac{599}{2352}\right)\) | \(e\left(\frac{1553}{4704}\right)\) | \(e\left(\frac{65}{1568}\right)\) | \(e\left(\frac{199}{392}\right)\) | \(e\left(\frac{179}{1176}\right)\) | \(e\left(\frac{91}{96}\right)\) | \(e\left(\frac{1931}{2352}\right)\) | \(e\left(\frac{1789}{2352}\right)\) |
| \(\chi_{43904}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{403}{4704}\right)\) | \(e\left(\frac{4337}{4704}\right)\) | \(e\left(\frac{403}{2352}\right)\) | \(e\left(\frac{2533}{4704}\right)\) | \(e\left(\frac{1045}{1568}\right)\) | \(e\left(\frac{3}{392}\right)\) | \(e\left(\frac{1159}{1176}\right)\) | \(e\left(\frac{23}{96}\right)\) | \(e\left(\frac{1735}{2352}\right)\) | \(e\left(\frac{1985}{2352}\right)\) |
| \(\chi_{43904}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4121}{4704}\right)\) | \(e\left(\frac{3379}{4704}\right)\) | \(e\left(\frac{1769}{2352}\right)\) | \(e\left(\frac{1775}{4704}\right)\) | \(e\left(\frac{671}{1568}\right)\) | \(e\left(\frac{233}{392}\right)\) | \(e\left(\frac{509}{1176}\right)\) | \(e\left(\frac{37}{96}\right)\) | \(e\left(\frac{1733}{2352}\right)\) | \(e\left(\frac{1027}{2352}\right)\) |
| \(\chi_{43904}(277,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{421}{4704}\right)\) | \(e\left(\frac{3095}{4704}\right)\) | \(e\left(\frac{421}{2352}\right)\) | \(e\left(\frac{3043}{4704}\right)\) | \(e\left(\frac{403}{1568}\right)\) | \(e\left(\frac{293}{392}\right)\) | \(e\left(\frac{169}{1176}\right)\) | \(e\left(\frac{65}{96}\right)\) | \(e\left(\frac{1153}{2352}\right)\) | \(e\left(\frac{743}{2352}\right)\) |
| \(\chi_{43904}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4283}{4704}\right)\) | \(e\left(\frac{1609}{4704}\right)\) | \(e\left(\frac{1931}{2352}\right)\) | \(e\left(\frac{1661}{4704}\right)\) | \(e\left(\frac{1165}{1568}\right)\) | \(e\left(\frac{99}{392}\right)\) | \(e\left(\frac{1007}{1176}\right)\) | \(e\left(\frac{31}{96}\right)\) | \(e\left(\frac{1199}{2352}\right)\) | \(e\left(\frac{1609}{2352}\right)\) |
| \(\chi_{43904}(333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{775}{4704}\right)\) | \(e\left(\frac{2189}{4704}\right)\) | \(e\left(\frac{775}{2352}\right)\) | \(e\left(\frac{529}{4704}\right)\) | \(e\left(\frac{321}{1568}\right)\) | \(e\left(\frac{247}{392}\right)\) | \(e\left(\frac{691}{1176}\right)\) | \(e\left(\frac{59}{96}\right)\) | \(e\left(\frac{2251}{2352}\right)\) | \(e\left(\frac{2189}{2352}\right)\) |
| \(\chi_{43904}(389,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1465}{4704}\right)\) | \(e\left(\frac{1619}{4704}\right)\) | \(e\left(\frac{1465}{2352}\right)\) | \(e\left(\frac{4399}{4704}\right)\) | \(e\left(\frac{799}{1568}\right)\) | \(e\left(\frac{257}{392}\right)\) | \(e\left(\frac{373}{1176}\right)\) | \(e\left(\frac{5}{96}\right)\) | \(e\left(\frac{325}{2352}\right)\) | \(e\left(\frac{1619}{2352}\right)\) |
| \(\chi_{43904}(429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1583}{4704}\right)\) | \(e\left(\frac{4453}{4704}\right)\) | \(e\left(\frac{1583}{2352}\right)\) | \(e\left(\frac{425}{4704}\right)\) | \(e\left(\frac{249}{1568}\right)\) | \(e\left(\frac{111}{392}\right)\) | \(e\left(\frac{155}{1176}\right)\) | \(e\left(\frac{67}{96}\right)\) | \(e\left(\frac{1475}{2352}\right)\) | \(e\left(\frac{2101}{2352}\right)\) |
| \(\chi_{43904}(445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2491}{4704}\right)\) | \(e\left(\frac{1385}{4704}\right)\) | \(e\left(\frac{139}{2352}\right)\) | \(e\left(\frac{541}{4704}\right)\) | \(e\left(\frac{269}{1568}\right)\) | \(e\left(\frac{323}{392}\right)\) | \(e\left(\frac{391}{1176}\right)\) | \(e\left(\frac{95}{96}\right)\) | \(e\left(\frac{79}{2352}\right)\) | \(e\left(\frac{1385}{2352}\right)\) |
| \(\chi_{43904}(485,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3425}{4704}\right)\) | \(e\left(\frac{4363}{4704}\right)\) | \(e\left(\frac{1073}{2352}\right)\) | \(e\left(\frac{4007}{4704}\right)\) | \(e\left(\frac{407}{1568}\right)\) | \(e\left(\frac{257}{392}\right)\) | \(e\left(\frac{1157}{1176}\right)\) | \(e\left(\frac{13}{96}\right)\) | \(e\left(\frac{2285}{2352}\right)\) | \(e\left(\frac{2011}{2352}\right)\) |
| \(\chi_{43904}(501,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3853}{4704}\right)\) | \(e\left(\frac{1487}{4704}\right)\) | \(e\left(\frac{1501}{2352}\right)\) | \(e\left(\frac{3067}{4704}\right)\) | \(e\left(\frac{299}{1568}\right)\) | \(e\left(\frac{53}{392}\right)\) | \(e\left(\frac{745}{1176}\right)\) | \(e\left(\frac{41}{96}\right)\) | \(e\left(\frac{1513}{2352}\right)\) | \(e\left(\frac{1487}{2352}\right)\) |
| \(\chi_{43904}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4259}{4704}\right)\) | \(e\left(\frac{3265}{4704}\right)\) | \(e\left(\frac{1907}{2352}\right)\) | \(e\left(\frac{2549}{4704}\right)\) | \(e\left(\frac{453}{1568}\right)\) | \(e\left(\frac{235}{392}\right)\) | \(e\left(\frac{1151}{1176}\right)\) | \(e\left(\frac{7}{96}\right)\) | \(e\left(\frac{407}{2352}\right)\) | \(e\left(\frac{913}{2352}\right)\) |
| \(\chi_{43904}(597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4085}{4704}\right)\) | \(e\left(\frac{1159}{4704}\right)\) | \(e\left(\frac{1733}{2352}\right)\) | \(e\left(\frac{755}{4704}\right)\) | \(e\left(\frac{387}{1568}\right)\) | \(e\left(\frac{45}{392}\right)\) | \(e\left(\frac{137}{1176}\right)\) | \(e\left(\frac{49}{96}\right)\) | \(e\left(\frac{545}{2352}\right)\) | \(e\left(\frac{1159}{2352}\right)\) |
| \(\chi_{43904}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2881}{4704}\right)\) | \(e\left(\frac{2699}{4704}\right)\) | \(e\left(\frac{529}{2352}\right)\) | \(e\left(\frac{3751}{4704}\right)\) | \(e\left(\frac{471}{1568}\right)\) | \(e\left(\frac{73}{392}\right)\) | \(e\left(\frac{109}{1176}\right)\) | \(e\left(\frac{77}{96}\right)\) | \(e\left(\frac{13}{2352}\right)\) | \(e\left(\frac{347}{2352}\right)\) |
| \(\chi_{43904}(653,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2903}{4704}\right)\) | \(e\left(\frac{2749}{4704}\right)\) | \(e\left(\frac{551}{2352}\right)\) | \(e\left(\frac{3329}{4704}\right)\) | \(e\left(\frac{209}{1568}\right)\) | \(e\left(\frac{79}{392}\right)\) | \(e\left(\frac{467}{1176}\right)\) | \(e\left(\frac{43}{96}\right)\) | \(e\left(\frac{347}{2352}\right)\) | \(e\left(\frac{397}{2352}\right)\) |
| \(\chi_{43904}(669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{547}{4704}\right)\) | \(e\left(\frac{3809}{4704}\right)\) | \(e\left(\frac{547}{2352}\right)\) | \(e\left(\frac{1909}{4704}\right)\) | \(e\left(\frac{613}{1568}\right)\) | \(e\left(\frac{363}{392}\right)\) | \(e\left(\frac{295}{1176}\right)\) | \(e\left(\frac{71}{96}\right)\) | \(e\left(\frac{1783}{2352}\right)\) | \(e\left(\frac{1457}{2352}\right)\) |
| \(\chi_{43904}(709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{713}{4704}\right)\) | \(e\left(\frac{3331}{4704}\right)\) | \(e\left(\frac{713}{2352}\right)\) | \(e\left(\frac{863}{4704}\right)\) | \(e\left(\frac{1487}{1568}\right)\) | \(e\left(\frac{337}{392}\right)\) | \(e\left(\frac{965}{1176}\right)\) | \(e\left(\frac{85}{96}\right)\) | \(e\left(\frac{2165}{2352}\right)\) | \(e\left(\frac{979}{2352}\right)\) |
| \(\chi_{43904}(725,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3253}{4704}\right)\) | \(e\left(\frac{551}{4704}\right)\) | \(e\left(\frac{901}{2352}\right)\) | \(e\left(\frac{1747}{4704}\right)\) | \(e\left(\frac{1315}{1568}\right)\) | \(e\left(\frac{317}{392}\right)\) | \(e\left(\frac{817}{1176}\right)\) | \(e\left(\frac{17}{96}\right)\) | \(e\left(\frac{529}{2352}\right)\) | \(e\left(\frac{551}{2352}\right)\) |
| \(\chi_{43904}(781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1591}{4704}\right)\) | \(e\left(\frac{2333}{4704}\right)\) | \(e\left(\frac{1591}{2352}\right)\) | \(e\left(\frac{3265}{4704}\right)\) | \(e\left(\frac{1009}{1568}\right)\) | \(e\left(\frac{327}{392}\right)\) | \(e\left(\frac{499}{1176}\right)\) | \(e\left(\frac{11}{96}\right)\) | \(e\left(\frac{955}{2352}\right)\) | \(e\left(\frac{2333}{2352}\right)\) |
| \(\chi_{43904}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2717}{4704}\right)\) | \(e\left(\frac{1471}{4704}\right)\) | \(e\left(\frac{365}{2352}\right)\) | \(e\left(\frac{4331}{4704}\right)\) | \(e\left(\frac{571}{1568}\right)\) | \(e\left(\frac{349}{392}\right)\) | \(e\left(\frac{113}{1176}\right)\) | \(e\left(\frac{25}{96}\right)\) | \(e\left(\frac{89}{2352}\right)\) | \(e\left(\frac{1471}{2352}\right)\) |
| \(\chi_{43904}(837,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{265}{4704}\right)\) | \(e\left(\frac{4451}{4704}\right)\) | \(e\left(\frac{265}{2352}\right)\) | \(e\left(\frac{1759}{4704}\right)\) | \(e\left(\frac{1263}{1568}\right)\) | \(e\left(\frac{1}{392}\right)\) | \(e\left(\frac{517}{1176}\right)\) | \(e\left(\frac{53}{96}\right)\) | \(e\left(\frac{709}{2352}\right)\) | \(e\left(\frac{2099}{2352}\right)\) |
| \(\chi_{43904}(877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2207}{4704}\right)\) | \(e\left(\frac{3733}{4704}\right)\) | \(e\left(\frac{2207}{2352}\right)\) | \(e\left(\frac{857}{4704}\right)\) | \(e\left(\frac{1513}{1568}\right)\) | \(e\left(\frac{103}{392}\right)\) | \(e\left(\frac{1115}{1176}\right)\) | \(e\left(\frac{19}{96}\right)\) | \(e\left(\frac{899}{2352}\right)\) | \(e\left(\frac{1381}{2352}\right)\) |
| \(\chi_{43904}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3979}{4704}\right)\) | \(e\left(\frac{2201}{4704}\right)\) | \(e\left(\frac{1627}{2352}\right)\) | \(e\left(\frac{1933}{4704}\right)\) | \(e\left(\frac{509}{1568}\right)\) | \(e\left(\frac{123}{392}\right)\) | \(e\left(\frac{871}{1176}\right)\) | \(e\left(\frac{47}{96}\right)\) | \(e\left(\frac{2143}{2352}\right)\) | \(e\left(\frac{2201}{2352}\right)\) |
| \(\chi_{43904}(933,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{689}{4704}\right)\) | \(e\left(\frac{283}{4704}\right)\) | \(e\left(\frac{689}{2352}\right)\) | \(e\left(\frac{1751}{4704}\right)\) | \(e\left(\frac{775}{1568}\right)\) | \(e\left(\frac{81}{392}\right)\) | \(e\left(\frac{1109}{1176}\right)\) | \(e\left(\frac{61}{96}\right)\) | \(e\left(\frac{1373}{2352}\right)\) | \(e\left(\frac{283}{2352}\right)\) |
| \(\chi_{43904}(989,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2867}{4704}\right)\) | \(e\left(\frac{529}{4704}\right)\) | \(e\left(\frac{515}{2352}\right)\) | \(e\left(\frac{2309}{4704}\right)\) | \(e\left(\frac{1493}{1568}\right)\) | \(e\left(\frac{283}{392}\right)\) | \(e\left(\frac{95}{1176}\right)\) | \(e\left(\frac{55}{96}\right)\) | \(e\left(\frac{1511}{2352}\right)\) | \(e\left(\frac{529}{2352}\right)\) |
| \(\chi_{43904}(1005,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3007}{4704}\right)\) | \(e\left(\frac{3413}{4704}\right)\) | \(e\left(\frac{655}{2352}\right)\) | \(e\left(\frac{2617}{4704}\right)\) | \(e\left(\frac{681}{1568}\right)\) | \(e\left(\frac{143}{392}\right)\) | \(e\left(\frac{235}{1176}\right)\) | \(e\left(\frac{83}{96}\right)\) | \(e\left(\frac{643}{2352}\right)\) | \(e\left(\frac{1061}{2352}\right)\) |