from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(790142, base_ring=CyclotomicField(2112))
M = H._module
chi = DirichletCharacter(H, M([864,864,407]))
chi.galois_orbit()
[g,chi] = znchar(Mod(57,790142))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(790142\) | |
| Conductor: | \(395071\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2112\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 395071.djr | 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: | $\Q(\zeta_{2112})$ |
| Fixed field: | Number field defined by a degree 2112 polynomial (not computed) |
First 31 of 640 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{790142}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{176}\right)\) | \(e\left(\frac{503}{2112}\right)\) | \(e\left(\frac{251}{264}\right)\) | \(e\left(\frac{25}{88}\right)\) | \(e\left(\frac{219}{704}\right)\) | \(e\left(\frac{217}{704}\right)\) | \(e\left(\frac{73}{192}\right)\) | \(e\left(\frac{617}{2112}\right)\) | \(e\left(\frac{839}{2112}\right)\) | \(e\left(\frac{49}{528}\right)\) |
| \(\chi_{790142}(845,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{176}\right)\) | \(e\left(\frac{137}{2112}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{229}{704}\right)\) | \(e\left(\frac{423}{704}\right)\) | \(e\left(\frac{55}{192}\right)\) | \(e\left(\frac{1751}{2112}\right)\) | \(e\left(\frac{1337}{2112}\right)\) | \(e\left(\frac{175}{528}\right)\) |
| \(\chi_{790142}(2873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{176}\right)\) | \(e\left(\frac{1235}{2112}\right)\) | \(e\left(\frac{167}{264}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{199}{704}\right)\) | \(e\left(\frac{509}{704}\right)\) | \(e\left(\frac{109}{192}\right)\) | \(e\left(\frac{461}{2112}\right)\) | \(e\left(\frac{1955}{2112}\right)\) | \(e\left(\frac{325}{528}\right)\) |
| \(\chi_{790142}(3469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{176}\right)\) | \(e\left(\frac{1931}{2112}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{607}{704}\right)\) | \(e\left(\frac{325}{704}\right)\) | \(e\left(\frac{181}{192}\right)\) | \(e\left(\frac{1109}{2112}\right)\) | \(e\left(\frac{731}{2112}\right)\) | \(e\left(\frac{397}{528}\right)\) |
| \(\chi_{790142}(4939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{176}\right)\) | \(e\left(\frac{2029}{2112}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{185}{704}\right)\) | \(e\left(\frac{643}{704}\right)\) | \(e\left(\frac{83}{192}\right)\) | \(e\left(\frac{1267}{2112}\right)\) | \(e\left(\frac{1117}{2112}\right)\) | \(e\left(\frac{395}{528}\right)\) |
| \(\chi_{790142}(5065,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{176}\right)\) | \(e\left(\frac{799}{2112}\right)\) | \(e\left(\frac{67}{264}\right)\) | \(e\left(\frac{81}{88}\right)\) | \(e\left(\frac{611}{704}\right)\) | \(e\left(\frac{689}{704}\right)\) | \(e\left(\frac{161}{192}\right)\) | \(e\left(\frac{577}{2112}\right)\) | \(e\left(\frac{367}{2112}\right)\) | \(e\left(\frac{377}{528}\right)\) |
| \(\chi_{790142}(6063,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{176}\right)\) | \(e\left(\frac{1027}{2112}\right)\) | \(e\left(\frac{175}{264}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{247}{704}\right)\) | \(e\left(\frac{653}{704}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{1117}{2112}\right)\) | \(e\left(\frac{403}{2112}\right)\) | \(e\left(\frac{437}{528}\right)\) |
| \(\chi_{790142}(6547,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{176}\right)\) | \(e\left(\frac{1703}{2112}\right)\) | \(e\left(\frac{83}{264}\right)\) | \(e\left(\frac{57}{88}\right)\) | \(e\left(\frac{267}{704}\right)\) | \(e\left(\frac{361}{704}\right)\) | \(e\left(\frac{25}{192}\right)\) | \(e\left(\frac{569}{2112}\right)\) | \(e\left(\frac{695}{2112}\right)\) | \(e\left(\frac{337}{528}\right)\) |
| \(\chi_{790142}(6967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{176}\right)\) | \(e\left(\frac{1499}{2112}\right)\) | \(e\left(\frac{167}{264}\right)\) | \(e\left(\frac{85}{88}\right)\) | \(e\left(\frac{111}{704}\right)\) | \(e\left(\frac{245}{704}\right)\) | \(e\left(\frac{37}{192}\right)\) | \(e\left(\frac{197}{2112}\right)\) | \(e\left(\frac{107}{2112}\right)\) | \(e\left(\frac{61}{528}\right)\) |
| \(\chi_{790142}(9033,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{176}\right)\) | \(e\left(\frac{49}{2112}\right)\) | \(e\left(\frac{205}{264}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{493}{704}\right)\) | \(e\left(\frac{511}{704}\right)\) | \(e\left(\frac{143}{192}\right)\) | \(e\left(\frac{1135}{2112}\right)\) | \(e\left(\frac{1249}{2112}\right)\) | \(e\left(\frac{263}{528}\right)\) |
| \(\chi_{790142}(9151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{141}{176}\right)\) | \(e\left(\frac{1795}{2112}\right)\) | \(e\left(\frac{247}{264}\right)\) | \(e\left(\frac{53}{88}\right)\) | \(e\left(\frac{503}{704}\right)\) | \(e\left(\frac{13}{704}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{157}{2112}\right)\) | \(e\left(\frac{1747}{2112}\right)\) | \(e\left(\frac{389}{528}\right)\) |
| \(\chi_{790142}(10641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{123}{176}\right)\) | \(e\left(\frac{1637}{2112}\right)\) | \(e\left(\frac{17}{264}\right)\) | \(e\left(\frac{35}{88}\right)\) | \(e\left(\frac{465}{704}\right)\) | \(e\left(\frac{75}{704}\right)\) | \(e\left(\frac{91}{192}\right)\) | \(e\left(\frac{635}{2112}\right)\) | \(e\left(\frac{1685}{2112}\right)\) | \(e\left(\frac{403}{528}\right)\) |
| \(\chi_{790142}(11657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{176}\right)\) | \(e\left(\frac{1909}{2112}\right)\) | \(e\left(\frac{169}{264}\right)\) | \(e\left(\frac{27}{88}\right)\) | \(e\left(\frac{673}{704}\right)\) | \(e\left(\frac{699}{704}\right)\) | \(e\left(\frac{11}{192}\right)\) | \(e\left(\frac{427}{2112}\right)\) | \(e\left(\frac{1765}{2112}\right)\) | \(e\left(\frac{419}{528}\right)\) |
| \(\chi_{790142}(17221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{176}\right)\) | \(e\left(\frac{1897}{2112}\right)\) | \(e\left(\frac{205}{264}\right)\) | \(e\left(\frac{39}{88}\right)\) | \(e\left(\frac{581}{704}\right)\) | \(e\left(\frac{71}{704}\right)\) | \(e\left(\frac{23}{192}\right)\) | \(e\left(\frac{1399}{2112}\right)\) | \(e\left(\frac{985}{2112}\right)\) | \(e\left(\frac{527}{528}\right)\) |
| \(\chi_{790142}(18345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{176}\right)\) | \(e\left(\frac{2017}{2112}\right)\) | \(e\left(\frac{109}{264}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{93}{704}\right)\) | \(e\left(\frac{15}{704}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{127}{2112}\right)\) | \(e\left(\frac{337}{2112}\right)\) | \(e\left(\frac{503}{528}\right)\) |
| \(\chi_{790142}(18801,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{176}\right)\) | \(e\left(\frac{835}{2112}\right)\) | \(e\left(\frac{223}{264}\right)\) | \(e\left(\frac{45}{88}\right)\) | \(e\left(\frac{183}{704}\right)\) | \(e\left(\frac{461}{704}\right)\) | \(e\left(\frac{125}{192}\right)\) | \(e\left(\frac{1885}{2112}\right)\) | \(e\left(\frac{595}{2112}\right)\) | \(e\left(\frac{53}{528}\right)\) |
| \(\chi_{790142}(19249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{176}\right)\) | \(e\left(\frac{113}{2112}\right)\) | \(e\left(\frac{101}{264}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{45}{704}\right)\) | \(e\left(\frac{575}{704}\right)\) | \(e\left(\frac{79}{192}\right)\) | \(e\left(\frac{1583}{2112}\right)\) | \(e\left(\frac{1889}{2112}\right)\) | \(e\left(\frac{391}{528}\right)\) |
| \(\chi_{790142}(19845,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{176}\right)\) | \(e\left(\frac{523}{2112}\right)\) | \(e\left(\frac{103}{264}\right)\) | \(e\left(\frac{5}{88}\right)\) | \(e\left(\frac{607}{704}\right)\) | \(e\left(\frac{325}{704}\right)\) | \(e\left(\frac{53}{192}\right)\) | \(e\left(\frac{1813}{2112}\right)\) | \(e\left(\frac{1435}{2112}\right)\) | \(e\left(\frac{221}{528}\right)\) |
| \(\chi_{790142}(21433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{176}\right)\) | \(e\left(\frac{673}{2112}\right)\) | \(e\left(\frac{181}{264}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{349}{704}\right)\) | \(e\left(\frac{79}{704}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{1279}{2112}\right)\) | \(e\left(\frac{1681}{2112}\right)\) | \(e\left(\frac{455}{528}\right)\) |
| \(\chi_{790142}(22439,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{95}{176}\right)\) | \(e\left(\frac{785}{2112}\right)\) | \(e\left(\frac{197}{264}\right)\) | \(e\left(\frac{7}{88}\right)\) | \(e\left(\frac{269}{704}\right)\) | \(e\left(\frac{543}{704}\right)\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{2063}{2112}\right)\) | \(e\left(\frac{1217}{2112}\right)\) | \(e\left(\frac{151}{528}\right)\) |
| \(\chi_{790142}(22923,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{176}\right)\) | \(e\left(\frac{1417}{2112}\right)\) | \(e\left(\frac{61}{264}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{421}{704}\right)\) | \(e\left(\frac{295}{704}\right)\) | \(e\left(\frac{119}{192}\right)\) | \(e\left(\frac{151}{2112}\right)\) | \(e\left(\frac{1465}{2112}\right)\) | \(e\left(\frac{95}{528}\right)\) |
| \(\chi_{790142}(23343,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{176}\right)\) | \(e\left(\frac{1345}{2112}\right)\) | \(e\left(\frac{13}{264}\right)\) | \(e\left(\frac{63}{88}\right)\) | \(e\left(\frac{573}{704}\right)\) | \(e\left(\frac{47}{704}\right)\) | \(e\left(\frac{191}{192}\right)\) | \(e\left(\frac{1759}{2112}\right)\) | \(e\left(\frac{1009}{2112}\right)\) | \(e\left(\frac{215}{528}\right)\) |
| \(\chi_{790142}(25527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{176}\right)\) | \(e\left(\frac{1553}{2112}\right)\) | \(e\left(\frac{5}{264}\right)\) | \(e\left(\frac{31}{88}\right)\) | \(e\left(\frac{525}{704}\right)\) | \(e\left(\frac{607}{704}\right)\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{1103}{2112}\right)\) | \(e\left(\frac{449}{2112}\right)\) | \(e\left(\frac{103}{528}\right)\) |
| \(\chi_{790142}(26963,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{176}\right)\) | \(e\left(\frac{23}{2112}\right)\) | \(e\left(\frac{107}{264}\right)\) | \(e\left(\frac{65}{88}\right)\) | \(e\left(\frac{59}{704}\right)\) | \(e\left(\frac{441}{704}\right)\) | \(e\left(\frac{169}{192}\right)\) | \(e\left(\frac{1481}{2112}\right)\) | \(e\left(\frac{1319}{2112}\right)\) | \(e\left(\frac{145}{528}\right)\) |
| \(\chi_{790142}(29503,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{176}\right)\) | \(e\left(\frac{5}{2112}\right)\) | \(e\left(\frac{161}{264}\right)\) | \(e\left(\frac{83}{88}\right)\) | \(e\left(\frac{625}{704}\right)\) | \(e\left(\frac{555}{704}\right)\) | \(e\left(\frac{187}{192}\right)\) | \(e\left(\frac{1883}{2112}\right)\) | \(e\left(\frac{1205}{2112}\right)\) | \(e\left(\frac{307}{528}\right)\) |
| \(\chi_{790142}(31083,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{176}\right)\) | \(e\left(\frac{1825}{2112}\right)\) | \(e\left(\frac{157}{264}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{29}{704}\right)\) | \(e\left(\frac{527}{704}\right)\) | \(e\left(\frac{95}{192}\right)\) | \(e\left(\frac{895}{2112}\right)\) | \(e\left(\frac{529}{2112}\right)\) | \(e\left(\frac{119}{528}\right)\) |
| \(\chi_{790142}(31111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{176}\right)\) | \(e\left(\frac{1681}{2112}\right)\) | \(e\left(\frac{61}{264}\right)\) | \(e\left(\frac{79}{88}\right)\) | \(e\left(\frac{333}{704}\right)\) | \(e\left(\frac{31}{704}\right)\) | \(e\left(\frac{47}{192}\right)\) | \(e\left(\frac{1999}{2112}\right)\) | \(e\left(\frac{1729}{2112}\right)\) | \(e\left(\frac{359}{528}\right)\) |
| \(\chi_{790142}(33597,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{105}{176}\right)\) | \(e\left(\frac{71}{2112}\right)\) | \(e\left(\frac{227}{264}\right)\) | \(e\left(\frac{17}{88}\right)\) | \(e\left(\frac{427}{704}\right)\) | \(e\left(\frac{137}{704}\right)\) | \(e\left(\frac{121}{192}\right)\) | \(e\left(\frac{1817}{2112}\right)\) | \(e\left(\frac{215}{2112}\right)\) | \(e\left(\frac{241}{528}\right)\) |
| \(\chi_{790142}(33723,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{176}\right)\) | \(e\left(\frac{1723}{2112}\right)\) | \(e\left(\frac{199}{264}\right)\) | \(e\left(\frac{37}{88}\right)\) | \(e\left(\frac{655}{704}\right)\) | \(e\left(\frac{469}{704}\right)\) | \(e\left(\frac{5}{192}\right)\) | \(e\left(\frac{1765}{2112}\right)\) | \(e\left(\frac{1291}{2112}\right)\) | \(e\left(\frac{509}{528}\right)\) |
| \(\chi_{790142}(34721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{117}{176}\right)\) | \(e\left(\frac{59}{2112}\right)\) | \(e\left(\frac{263}{264}\right)\) | \(e\left(\frac{29}{88}\right)\) | \(e\left(\frac{335}{704}\right)\) | \(e\left(\frac{213}{704}\right)\) | \(e\left(\frac{133}{192}\right)\) | \(e\left(\frac{677}{2112}\right)\) | \(e\left(\frac{1547}{2112}\right)\) | \(e\left(\frac{349}{528}\right)\) |
| \(\chi_{790142}(35177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{176}\right)\) | \(e\left(\frac{593}{2112}\right)\) | \(e\left(\frac{245}{264}\right)\) | \(e\left(\frac{23}{88}\right)\) | \(e\left(\frac{205}{704}\right)\) | \(e\left(\frac{351}{704}\right)\) | \(e\left(\frac{175}{192}\right)\) | \(e\left(\frac{719}{2112}\right)\) | \(e\left(\frac{1409}{2112}\right)\) | \(e\left(\frac{295}{528}\right)\) |