from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(260))
M = H._module
chi = DirichletCharacter(H, M([104,15]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,1859))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1859\) | |
| Conductor: | \(1859\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(260\) | 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: | $\Q(\zeta_{260})$ |
| Fixed field: | Number field defined by a degree 260 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1859}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{260}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{119}{130}\right)\) | \(e\left(\frac{31}{260}\right)\) | \(e\left(\frac{211}{260}\right)\) | \(e\left(\frac{253}{260}\right)\) | \(e\left(\frac{97}{260}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
| \(\chi_{1859}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{260}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{159}{260}\right)\) | \(e\left(\frac{59}{260}\right)\) | \(e\left(\frac{157}{260}\right)\) | \(e\left(\frac{53}{260}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) |
| \(\chi_{1859}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{260}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{53}{130}\right)\) | \(e\left(\frac{217}{260}\right)\) | \(e\left(\frac{177}{260}\right)\) | \(e\left(\frac{211}{260}\right)\) | \(e\left(\frac{159}{260}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
| \(\chi_{1859}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{260}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{101}{260}\right)\) | \(e\left(\frac{201}{260}\right)\) | \(e\left(\frac{103}{260}\right)\) | \(e\left(\frac{207}{260}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) |
| \(\chi_{1859}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{260}\right)\) | \(e\left(\frac{37}{65}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{129}{260}\right)\) | \(e\left(\frac{249}{260}\right)\) | \(e\left(\frac{147}{260}\right)\) | \(e\left(\frac{43}{260}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) |
| \(\chi_{1859}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{260}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{93}{260}\right)\) | \(e\left(\frac{113}{260}\right)\) | \(e\left(\frac{239}{260}\right)\) | \(e\left(\frac{31}{260}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
| \(\chi_{1859}(135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{260}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{47}{260}\right)\) | \(e\left(\frac{127}{260}\right)\) | \(e\left(\frac{241}{260}\right)\) | \(e\left(\frac{189}{260}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) |
| \(\chi_{1859}(148,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{99}{260}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{111}{260}\right)\) | \(e\left(\frac{51}{260}\right)\) | \(e\left(\frac{193}{260}\right)\) | \(e\left(\frac{37}{260}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |
| \(\chi_{1859}(174,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{260}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{41}{130}\right)\) | \(e\left(\frac{239}{260}\right)\) | \(e\left(\frac{159}{260}\right)\) | \(e\left(\frac{97}{260}\right)\) | \(e\left(\frac{253}{260}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
| \(\chi_{1859}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{260}\right)\) | \(e\left(\frac{1}{65}\right)\) | \(e\left(\frac{73}{130}\right)\) | \(e\left(\frac{137}{260}\right)\) | \(e\left(\frac{77}{260}\right)\) | \(e\left(\frac{11}{260}\right)\) | \(e\left(\frac{219}{260}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |
| \(\chi_{1859}(203,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{260}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{89}{130}\right)\) | \(e\left(\frac{21}{260}\right)\) | \(e\left(\frac{101}{260}\right)\) | \(e\left(\frac{163}{260}\right)\) | \(e\left(\frac{7}{260}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) |
| \(\chi_{1859}(213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{260}\right)\) | \(e\left(\frac{4}{65}\right)\) | \(e\left(\frac{97}{130}\right)\) | \(e\left(\frac{223}{260}\right)\) | \(e\left(\frac{243}{260}\right)\) | \(e\left(\frac{109}{260}\right)\) | \(e\left(\frac{161}{260}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{21}{26}\right)\) |
| \(\chi_{1859}(229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{260}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{49}{260}\right)\) | \(e\left(\frac{149}{260}\right)\) | \(e\left(\frac{207}{260}\right)\) | \(e\left(\frac{103}{260}\right)\) | \(e\left(\frac{14}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) |
| \(\chi_{1859}(278,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{260}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{127}{260}\right)\) | \(e\left(\frac{227}{260}\right)\) | \(e\left(\frac{181}{260}\right)\) | \(e\left(\frac{129}{260}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) |
| \(\chi_{1859}(291,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{260}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{191}{260}\right)\) | \(e\left(\frac{151}{260}\right)\) | \(e\left(\frac{133}{260}\right)\) | \(e\left(\frac{237}{260}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
| \(\chi_{1859}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{260}\right)\) | \(e\left(\frac{27}{65}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{59}{260}\right)\) | \(e\left(\frac{259}{260}\right)\) | \(e\left(\frac{37}{260}\right)\) | \(e\left(\frac{193}{260}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) |
| \(\chi_{1859}(333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{260}\right)\) | \(e\left(\frac{36}{65}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{57}{260}\right)\) | \(e\left(\frac{237}{260}\right)\) | \(e\left(\frac{71}{260}\right)\) | \(e\left(\frac{19}{260}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) |
| \(\chi_{1859}(346,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{260}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{109}{130}\right)\) | \(e\left(\frac{201}{260}\right)\) | \(e\left(\frac{1}{260}\right)\) | \(e\left(\frac{223}{260}\right)\) | \(e\left(\frac{67}{260}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
| \(\chi_{1859}(356,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{260}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{77}{130}\right)\) | \(e\left(\frac{43}{260}\right)\) | \(e\left(\frac{83}{260}\right)\) | \(e\left(\frac{49}{260}\right)\) | \(e\left(\frac{101}{260}\right)\) | \(e\left(\frac{3}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) |
| \(\chi_{1859}(372,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{260}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{11}{130}\right)\) | \(e\left(\frac{229}{260}\right)\) | \(e\left(\frac{49}{260}\right)\) | \(e\left(\frac{7}{260}\right)\) | \(e\left(\frac{163}{260}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) |
| \(\chi_{1859}(411,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{260}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{193}{260}\right)\) | \(e\left(\frac{173}{260}\right)\) | \(e\left(\frac{99}{260}\right)\) | \(e\left(\frac{151}{260}\right)\) | \(e\left(\frac{18}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) |
| \(\chi_{1859}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{260}\right)\) | \(e\left(\frac{11}{65}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{207}{260}\right)\) | \(e\left(\frac{67}{260}\right)\) | \(e\left(\frac{121}{260}\right)\) | \(e\left(\frac{69}{260}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) |
| \(\chi_{1859}(434,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{260}\right)\) | \(e\left(\frac{48}{65}\right)\) | \(e\left(\frac{59}{130}\right)\) | \(e\left(\frac{11}{260}\right)\) | \(e\left(\frac{251}{260}\right)\) | \(e\left(\frac{73}{260}\right)\) | \(e\left(\frac{177}{260}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) |
| \(\chi_{1859}(460,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{260}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{139}{260}\right)\) | \(e\left(\frac{99}{260}\right)\) | \(e\left(\frac{237}{260}\right)\) | \(e\left(\frac{133}{260}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
| \(\chi_{1859}(476,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{260}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{237}{260}\right)\) | \(e\left(\frac{137}{260}\right)\) | \(e\left(\frac{131}{260}\right)\) | \(e\left(\frac{79}{260}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) |
| \(\chi_{1859}(489,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{129}{260}\right)\) | \(e\left(\frac{8}{65}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{121}{260}\right)\) | \(e\left(\frac{161}{260}\right)\) | \(e\left(\frac{23}{260}\right)\) | \(e\left(\frac{127}{260}\right)\) | \(e\left(\frac{16}{65}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{3}{26}\right)\) |
| \(\chi_{1859}(499,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{260}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{123}{260}\right)\) | \(e\left(\frac{183}{260}\right)\) | \(e\left(\frac{249}{260}\right)\) | \(e\left(\frac{41}{260}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
| \(\chi_{1859}(515,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{260}\right)\) | \(e\left(\frac{12}{65}\right)\) | \(e\left(\frac{31}{130}\right)\) | \(e\left(\frac{149}{260}\right)\) | \(e\left(\frac{209}{260}\right)\) | \(e\left(\frac{67}{260}\right)\) | \(e\left(\frac{223}{260}\right)\) | \(e\left(\frac{24}{65}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) |
| \(\chi_{1859}(554,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{260}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{27}{130}\right)\) | \(e\left(\frac{113}{260}\right)\) | \(e\left(\frac{73}{260}\right)\) | \(e\left(\frac{159}{260}\right)\) | \(e\left(\frac{211}{260}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{23}{26}\right)\) |
| \(\chi_{1859}(564,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{260}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{3}{130}\right)\) | \(e\left(\frac{27}{260}\right)\) | \(e\left(\frac{167}{260}\right)\) | \(e\left(\frac{61}{260}\right)\) | \(e\left(\frac{9}{260}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{17}{26}\right)\) |
| \(\chi_{1859}(603,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{260}\right)\) | \(e\left(\frac{22}{65}\right)\) | \(e\left(\frac{111}{130}\right)\) | \(e\left(\frac{219}{260}\right)\) | \(e\left(\frac{199}{260}\right)\) | \(e\left(\frac{177}{260}\right)\) | \(e\left(\frac{73}{260}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{26}\right)\) |