from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(327184, base_ring=CyclotomicField(2860))
M = H._module
chi = DirichletCharacter(H, M([0,715,2756,2200]))
chi.galois_orbit()
[g,chi] = znchar(Mod(53,327184))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(327184\) | |
| Conductor: | \(327184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2860\) | 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: | $\Q(\zeta_{2860})$ |
| Fixed field: | Number field defined by a degree 2860 polynomial (not computed) |
First 31 of 960 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{327184}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{243}{260}\right)\) | \(e\left(\frac{1379}{2860}\right)\) | \(e\left(\frac{791}{1430}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{298}{715}\right)\) | \(e\left(\frac{376}{715}\right)\) | \(e\left(\frac{161}{220}\right)\) | \(e\left(\frac{279}{572}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1379}{1430}\right)\) |
| \(\chi_{327184}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{260}\right)\) | \(e\left(\frac{757}{2860}\right)\) | \(e\left(\frac{1073}{1430}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{49}{715}\right)\) | \(e\left(\frac{153}{715}\right)\) | \(e\left(\frac{123}{220}\right)\) | \(e\left(\frac{317}{572}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{757}{1430}\right)\) |
| \(\chi_{327184}(885,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{260}\right)\) | \(e\left(\frac{1291}{2860}\right)\) | \(e\left(\frac{109}{1430}\right)\) | \(e\left(\frac{127}{130}\right)\) | \(e\left(\frac{672}{715}\right)\) | \(e\left(\frac{464}{715}\right)\) | \(e\left(\frac{29}{220}\right)\) | \(e\left(\frac{323}{572}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1291}{1430}\right)\) |
| \(\chi_{327184}(1093,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{260}\right)\) | \(e\left(\frac{2803}{2860}\right)\) | \(e\left(\frac{127}{1430}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{291}{715}\right)\) | \(e\left(\frac{252}{715}\right)\) | \(e\left(\frac{57}{220}\right)\) | \(e\left(\frac{295}{572}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1373}{1430}\right)\) |
| \(\chi_{327184}(1197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{233}{260}\right)\) | \(e\left(\frac{2649}{2860}\right)\) | \(e\left(\frac{721}{1430}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{588}{715}\right)\) | \(e\left(\frac{406}{715}\right)\) | \(e\left(\frac{211}{220}\right)\) | \(e\left(\frac{229}{572}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1219}{1430}\right)\) |
| \(\chi_{327184}(1301,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{260}\right)\) | \(e\left(\frac{2807}{2860}\right)\) | \(e\left(\frac{93}{1430}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{534}{715}\right)\) | \(e\left(\frac{573}{715}\right)\) | \(e\left(\frac{53}{220}\right)\) | \(e\left(\frac{475}{572}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1377}{1430}\right)\) |
| \(\chi_{327184}(2237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{260}\right)\) | \(e\left(\frac{2773}{2860}\right)\) | \(e\left(\frac{1097}{1430}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{256}{715}\right)\) | \(e\left(\frac{347}{715}\right)\) | \(e\left(\frac{87}{220}\right)\) | \(e\left(\frac{89}{572}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1343}{1430}\right)\) |
| \(\chi_{327184}(2341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{260}\right)\) | \(e\left(\frac{1059}{2860}\right)\) | \(e\left(\frac{651}{1430}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{163}{715}\right)\) | \(e\left(\frac{436}{715}\right)\) | \(e\left(\frac{41}{220}\right)\) | \(e\left(\frac{179}{572}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1059}{1430}\right)\) |
| \(\chi_{327184}(2445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{189}{260}\right)\) | \(e\left(\frac{1997}{2860}\right)\) | \(e\left(\frac{543}{1430}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{304}{715}\right)\) | \(e\left(\frac{278}{715}\right)\) | \(e\left(\frac{203}{220}\right)\) | \(e\left(\frac{61}{572}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{567}{1430}\right)\) |
| \(\chi_{327184}(3485,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{260}\right)\) | \(e\left(\frac{2329}{2860}\right)\) | \(e\left(\frac{581}{1430}\right)\) | \(e\left(\frac{83}{130}\right)\) | \(e\left(\frac{453}{715}\right)\) | \(e\left(\frac{466}{715}\right)\) | \(e\left(\frac{91}{220}\right)\) | \(e\left(\frac{129}{572}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{899}{1430}\right)\) |
| \(\chi_{327184}(3589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{260}\right)\) | \(e\left(\frac{1187}{2860}\right)\) | \(e\left(\frac{993}{1430}\right)\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{74}{715}\right)\) | \(e\left(\frac{698}{715}\right)\) | \(e\left(\frac{133}{220}\right)\) | \(e\left(\frac{219}{572}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1187}{1430}\right)\) |
| \(\chi_{327184}(4317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{260}\right)\) | \(e\left(\frac{1461}{2860}\right)\) | \(e\left(\frac{809}{1430}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{632}{715}\right)\) | \(e\left(\frac{164}{715}\right)\) | \(e\left(\frac{79}{220}\right)\) | \(e\left(\frac{537}{572}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{31}{1430}\right)\) |
| \(\chi_{327184}(4525,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{260}\right)\) | \(e\left(\frac{2713}{2860}\right)\) | \(e\left(\frac{177}{1430}\right)\) | \(e\left(\frac{81}{130}\right)\) | \(e\left(\frac{186}{715}\right)\) | \(e\left(\frac{537}{715}\right)\) | \(e\left(\frac{147}{220}\right)\) | \(e\left(\frac{249}{572}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1283}{1430}\right)\) |
| \(\chi_{327184}(4629,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{203}{260}\right)\) | \(e\left(\frac{739}{2860}\right)\) | \(e\left(\frac{511}{1430}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{28}{715}\right)\) | \(e\left(\frac{496}{715}\right)\) | \(e\left(\frac{141}{220}\right)\) | \(e\left(\frac{79}{572}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{739}{1430}\right)\) |
| \(\chi_{327184}(5461,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{260}\right)\) | \(e\left(\frac{2471}{2860}\right)\) | \(e\left(\frac{89}{1430}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{142}{715}\right)\) | \(e\left(\frac{64}{715}\right)\) | \(e\left(\frac{169}{220}\right)\) | \(e\left(\frac{227}{572}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{1041}{1430}\right)\) |
| \(\chi_{327184}(5669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{260}\right)\) | \(e\left(\frac{2683}{2860}\right)\) | \(e\left(\frac{1147}{1430}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{151}{715}\right)\) | \(e\left(\frac{632}{715}\right)\) | \(e\left(\frac{177}{220}\right)\) | \(e\left(\frac{43}{572}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1253}{1430}\right)\) |
| \(\chi_{327184}(5773,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{260}\right)\) | \(e\left(\frac{2009}{2860}\right)\) | \(e\left(\frac{441}{1430}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{318}{715}\right)\) | \(e\left(\frac{526}{715}\right)\) | \(e\left(\frac{191}{220}\right)\) | \(e\left(\frac{29}{572}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{579}{1430}\right)\) |
| \(\chi_{327184}(5877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{260}\right)\) | \(e\left(\frac{2427}{2860}\right)\) | \(e\left(\frac{463}{1430}\right)\) | \(e\left(\frac{29}{130}\right)\) | \(e\left(\frac{329}{715}\right)\) | \(e\left(\frac{108}{715}\right)\) | \(e\left(\frac{213}{220}\right)\) | \(e\left(\frac{535}{572}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{997}{1430}\right)\) |
| \(\chi_{327184}(6605,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{260}\right)\) | \(e\left(\frac{621}{2860}\right)\) | \(e\left(\frac{799}{1430}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{367}{715}\right)\) | \(e\left(\frac{679}{715}\right)\) | \(e\left(\frac{39}{220}\right)\) | \(e\left(\frac{489}{572}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{621}{1430}\right)\) |
| \(\chi_{327184}(6813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{260}\right)\) | \(e\left(\frac{2653}{2860}\right)\) | \(e\left(\frac{687}{1430}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{116}{715}\right)\) | \(e\left(\frac{12}{715}\right)\) | \(e\left(\frac{207}{220}\right)\) | \(e\left(\frac{409}{572}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1223}{1430}\right)\) |
| \(\chi_{327184}(6917,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{183}{260}\right)\) | \(e\left(\frac{419}{2860}\right)\) | \(e\left(\frac{371}{1430}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{608}{715}\right)\) | \(e\left(\frac{556}{715}\right)\) | \(e\left(\frac{21}{220}\right)\) | \(e\left(\frac{551}{572}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{419}{1430}\right)\) |
| \(\chi_{327184}(7749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{260}\right)\) | \(e\left(\frac{1631}{2860}\right)\) | \(e\left(\frac{79}{1430}\right)\) | \(e\left(\frac{67}{130}\right)\) | \(e\left(\frac{592}{715}\right)\) | \(e\left(\frac{579}{715}\right)\) | \(e\left(\frac{129}{220}\right)\) | \(e\left(\frac{179}{572}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{201}{1430}\right)\) |
| \(\chi_{327184}(7957,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{260}\right)\) | \(e\left(\frac{2623}{2860}\right)\) | \(e\left(\frac{227}{1430}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{81}{715}\right)\) | \(e\left(\frac{107}{715}\right)\) | \(e\left(\frac{17}{220}\right)\) | \(e\left(\frac{203}{572}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1193}{1430}\right)\) |
| \(\chi_{327184}(8061,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{260}\right)\) | \(e\left(\frac{1689}{2860}\right)\) | \(e\left(\frac{301}{1430}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{183}{715}\right)\) | \(e\left(\frac{586}{715}\right)\) | \(e\left(\frac{71}{220}\right)\) | \(e\left(\frac{501}{572}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{259}{1430}\right)\) |
| \(\chi_{327184}(8165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{260}\right)\) | \(e\left(\frac{807}{2860}\right)\) | \(e\left(\frac{1363}{1430}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{584}{715}\right)\) | \(e\left(\frac{233}{715}\right)\) | \(e\left(\frac{73}{220}\right)\) | \(e\left(\frac{279}{572}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{807}{1430}\right)\) |
| \(\chi_{327184}(8893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{260}\right)\) | \(e\left(\frac{2641}{2860}\right)\) | \(e\left(\frac{789}{1430}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{102}{715}\right)\) | \(e\left(\frac{479}{715}\right)\) | \(e\left(\frac{219}{220}\right)\) | \(e\left(\frac{441}{572}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{1211}{1430}\right)\) |
| \(\chi_{327184}(9101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{260}\right)\) | \(e\left(\frac{2593}{2860}\right)\) | \(e\left(\frac{1197}{1430}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{46}{715}\right)\) | \(e\left(\frac{202}{715}\right)\) | \(e\left(\frac{47}{220}\right)\) | \(e\left(\frac{569}{572}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1163}{1430}\right)\) |
| \(\chi_{327184}(9309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{260}\right)\) | \(e\left(\frac{2857}{2860}\right)\) | \(e\left(\frac{383}{1430}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{354}{715}\right)\) | \(e\left(\frac{653}{715}\right)\) | \(e\left(\frac{3}{220}\right)\) | \(e\left(\frac{437}{572}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1427}{1430}\right)\) |
| \(\chi_{327184}(10037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{260}\right)\) | \(e\left(\frac{791}{2860}\right)\) | \(e\left(\frac{69}{1430}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{327}{715}\right)\) | \(e\left(\frac{379}{715}\right)\) | \(e\left(\frac{89}{220}\right)\) | \(e\left(\frac{131}{572}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{791}{1430}\right)\) |
| \(\chi_{327184}(10349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{153}{260}\right)\) | \(e\left(\frac{1369}{2860}\right)\) | \(e\left(\frac{161}{1430}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{48}{715}\right)\) | \(e\left(\frac{646}{715}\right)\) | \(e\left(\frac{171}{220}\right)\) | \(e\left(\frac{401}{572}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1369}{1430}\right)\) |
| \(\chi_{327184}(10453,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{260}\right)\) | \(e\left(\frac{2047}{2860}\right)\) | \(e\left(\frac{833}{1430}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{124}{715}\right)\) | \(e\left(\frac{358}{715}\right)\) | \(e\left(\frac{153}{220}\right)\) | \(e\left(\frac{23}{572}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{617}{1430}\right)\) |