from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1137, base_ring=CyclotomicField(378))
M = H._module
chi = DirichletCharacter(H, M([0,155]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,1137))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1137\) | |
| Conductor: | \(379\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(378\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 379.p | 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: | $\Q(\zeta_{189})$ |
| Fixed field: | Number field defined by a degree 378 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1137}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{378}\right)\) | \(e\left(\frac{155}{189}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{211}{378}\right)\) | \(e\left(\frac{29}{126}\right)\) | \(e\left(\frac{173}{378}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{131}{378}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{121}{189}\right)\) |
| \(\chi_{1137}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{378}\right)\) | \(e\left(\frac{145}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{173}{378}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{235}{378}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{25}{378}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{101}{189}\right)\) |
| \(\chi_{1137}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{378}\right)\) | \(e\left(\frac{124}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{131}{378}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{25}{378}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{67}{378}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{59}{189}\right)\) |
| \(\chi_{1137}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{378}\right)\) | \(e\left(\frac{157}{189}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{143}{378}\right)\) | \(e\left(\frac{31}{126}\right)\) | \(e\left(\frac{85}{378}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{1}{378}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{125}{189}\right)\) |
| \(\chi_{1137}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{205}{378}\right)\) | \(e\left(\frac{16}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{23}{378}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{241}{378}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{283}{378}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{32}{189}\right)\) |
| \(\chi_{1137}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{378}\right)\) | \(e\left(\frac{107}{189}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{331}{378}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{17}{378}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{227}{378}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{25}{189}\right)\) |
| \(\chi_{1137}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{378}\right)\) | \(e\left(\frac{25}{189}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{95}{378}\right)\) | \(e\left(\frac{25}{126}\right)\) | \(e\left(\frac{223}{378}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{265}{378}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{50}{189}\right)\) |
| \(\chi_{1137}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{378}\right)\) | \(e\left(\frac{172}{189}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{11}{378}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{181}{378}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{349}{378}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{155}{189}\right)\) |
| \(\chi_{1137}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{378}\right)\) | \(e\left(\frac{61}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{378}\right)\) | \(e\left(\frac{61}{126}\right)\) | \(e\left(\frac{151}{378}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{193}{378}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{122}{189}\right)\) |
| \(\chi_{1137}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{378}\right)\) | \(e\left(\frac{178}{189}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{185}{378}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{295}{378}\right)\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{337}{378}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{167}{189}\right)\) |
| \(\chi_{1137}(154,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{378}\right)\) | \(e\left(\frac{184}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{359}{378}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{31}{378}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{325}{378}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{179}{189}\right)\) |
| \(\chi_{1137}(160,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{378}\right)\) | \(e\left(\frac{149}{189}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{37}{378}\right)\) | \(e\left(\frac{23}{126}\right)\) | \(e\left(\frac{59}{378}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{143}{378}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{109}{189}\right)\) |
| \(\chi_{1137}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{378}\right)\) | \(e\left(\frac{109}{189}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{263}{378}\right)\) | \(e\left(\frac{109}{126}\right)\) | \(e\left(\frac{307}{378}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{97}{378}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{29}{189}\right)\) |
| \(\chi_{1137}(175,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{378}\right)\) | \(e\left(\frac{65}{189}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{247}{378}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{353}{378}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{311}{378}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{130}{189}\right)\) |
| \(\chi_{1137}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{377}{378}\right)\) | \(e\left(\frac{188}{189}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{223}{378}\right)\) | \(e\left(\frac{125}{126}\right)\) | \(e\left(\frac{233}{378}\right)\) | \(e\left(\frac{23}{54}\right)\) | \(e\left(\frac{65}{378}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{187}{189}\right)\) |
| \(\chi_{1137}(208,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{378}\right)\) | \(e\left(\frac{128}{189}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{373}{378}\right)\) | \(e\left(\frac{65}{126}\right)\) | \(e\left(\frac{227}{378}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{185}{378}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{67}{189}\right)\) |
| \(\chi_{1137}(223,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{263}{378}\right)\) | \(e\left(\frac{74}{189}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{319}{378}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{335}{378}\right)\) | \(e\left(\frac{53}{54}\right)\) | \(e\left(\frac{293}{378}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{148}{189}\right)\) |
| \(\chi_{1137}(235,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{378}\right)\) | \(e\left(\frac{89}{189}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{187}{378}\right)\) | \(e\left(\frac{89}{126}\right)\) | \(e\left(\frac{53}{378}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{263}{378}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{178}{189}\right)\) |
| \(\chi_{1137}(238,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{378}\right)\) | \(e\left(\frac{82}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{47}{378}\right)\) | \(e\left(\frac{19}{126}\right)\) | \(e\left(\frac{361}{378}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{151}{378}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{164}{189}\right)\) |
| \(\chi_{1137}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{378}\right)\) | \(e\left(\frac{167}{189}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{181}{378}\right)\) | \(e\left(\frac{41}{126}\right)\) | \(e\left(\frac{23}{378}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{107}{378}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{145}{189}\right)\) |
| \(\chi_{1137}(250,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{378}\right)\) | \(e\left(\frac{55}{189}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{209}{378}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{37}{378}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{205}{378}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{110}{189}\right)\) |
| \(\chi_{1137}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{241}{378}\right)\) | \(e\left(\frac{52}{189}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{311}{378}\right)\) | \(e\left(\frac{115}{126}\right)\) | \(e\left(\frac{169}{378}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{211}{378}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{104}{189}\right)\) |
| \(\chi_{1137}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{378}\right)\) | \(e\left(\frac{95}{189}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{361}{378}\right)\) | \(e\left(\frac{95}{126}\right)\) | \(e\left(\frac{167}{378}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{251}{378}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{1}{189}\right)\) |
| \(\chi_{1137}(265,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{378}\right)\) | \(e\left(\frac{181}{189}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{83}{378}\right)\) | \(e\left(\frac{55}{126}\right)\) | \(e\left(\frac{163}{378}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{331}{378}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{173}{189}\right)\) |
| \(\chi_{1137}(274,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{247}{378}\right)\) | \(e\left(\frac{58}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{107}{378}\right)\) | \(e\left(\frac{121}{126}\right)\) | \(e\left(\frac{283}{378}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{199}{378}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{116}{189}\right)\) |
| \(\chi_{1137}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{378}\right)\) | \(e\left(\frac{142}{189}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{275}{378}\right)\) | \(e\left(\frac{79}{126}\right)\) | \(e\left(\frac{367}{378}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{31}{378}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{95}{189}\right)\) |
| \(\chi_{1137}(298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{378}\right)\) | \(e\left(\frac{170}{189}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{79}{378}\right)\) | \(e\left(\frac{107}{126}\right)\) | \(e\left(\frac{269}{378}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{101}{378}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{151}{189}\right)\) |
| \(\chi_{1137}(325,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{378}\right)\) | \(e\left(\frac{34}{189}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{167}{378}\right)\) | \(e\left(\frac{97}{126}\right)\) | \(e\left(\frac{205}{378}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{247}{378}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{68}{189}\right)\) |
| \(\chi_{1137}(334,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{378}\right)\) | \(e\left(\frac{40}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{341}{378}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{319}{378}\right)\) | \(e\left(\frac{25}{54}\right)\) | \(e\left(\frac{235}{378}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{80}{189}\right)\) |
| \(\chi_{1137}(358,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{378}\right)\) | \(e\left(\frac{103}{189}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{89}{378}\right)\) | \(e\left(\frac{103}{126}\right)\) | \(e\left(\frac{193}{378}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{109}{378}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{17}{189}\right)\) |
| \(\chi_{1137}(370,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{378}\right)\) | \(e\left(\frac{85}{189}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{323}{378}\right)\) | \(e\left(\frac{85}{126}\right)\) | \(e\left(\frac{229}{378}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{145}{378}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{170}{189}\right)\) |