from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(162240, base_ring=CyclotomicField(312))
M = H._module
chi = DirichletCharacter(H, M([156,117,156,156,194]))
chi.galois_orbit()
[g,chi] = znchar(Mod(119,162240))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(162240\) | |
| Conductor: | \(81120\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(312\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 81120.bfw | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{312})$ |
| Fixed field: | Number field defined by a degree 312 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{162240}(119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{287}{312}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{155}{312}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{239}{312}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{229}{312}\right)\) |
| \(\chi_{162240}(839,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{49}{312}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{133}{312}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{193}{312}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{275}{312}\right)\) |
| \(\chi_{162240}(4439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{19}{312}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{312}\right)\) | \(e\left(\frac{5}{52}\right)\) | \(e\left(\frac{43}{312}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{113}{312}\right)\) |
| \(\chi_{162240}(6359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{83}{312}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{47}{312}\right)\) | \(e\left(\frac{41}{52}\right)\) | \(e\left(\frac{155}{312}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{1}{312}\right)\) |
| \(\chi_{162240}(10679,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{55}{312}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{283}{312}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{223}{312}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{245}{312}\right)\) |
| \(\chi_{162240}(11399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{113}{312}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{173}{312}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{305}{312}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{163}{312}\right)\) |
| \(\chi_{162240}(12599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{191}{312}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{251}{312}\right)\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{71}{312}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{85}{312}\right)\) |
| \(\chi_{162240}(13319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{289}{312}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{205}{312}\right)\) | \(e\left(\frac{35}{52}\right)\) | \(e\left(\frac{145}{312}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{11}{312}\right)\) |
| \(\chi_{162240}(17639,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{5}{312}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{281}{312}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{77}{312}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{79}{312}\right)\) |
| \(\chi_{162240}(19559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{253}{312}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{241}{312}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{277}{312}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{191}{312}\right)\) |
| \(\chi_{162240}(23159,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{127}{312}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{211}{312}\right)\) | \(e\left(\frac{17}{52}\right)\) | \(e\left(\frac{271}{312}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{197}{312}\right)\) |
| \(\chi_{162240}(23879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{209}{312}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{77}{312}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{161}{312}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{307}{312}\right)\) |
| \(\chi_{162240}(25079,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{95}{312}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{35}{312}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{215}{312}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{253}{312}\right)\) |
| \(\chi_{162240}(25799,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{217}{312}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{277}{312}\right)\) | \(e\left(\frac{27}{52}\right)\) | \(e\left(\frac{97}{312}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{59}{312}\right)\) |
| \(\chi_{162240}(29399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{163}{312}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{175}{312}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{139}{312}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{17}{312}\right)\) |
| \(\chi_{162240}(30119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{101}{312}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{185}{312}\right)\) | \(e\left(\frac{43}{52}\right)\) | \(e\left(\frac{245}{312}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{223}{312}\right)\) |
| \(\chi_{162240}(31319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{203}{312}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{239}{312}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{131}{312}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{25}{312}\right)\) |
| \(\chi_{162240}(32039,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{181}{312}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{312}\right)\) | \(e\left(\frac{23}{52}\right)\) | \(e\left(\frac{229}{312}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{239}{312}\right)\) |
| \(\chi_{162240}(35639,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{199}{312}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{139}{312}\right)\) | \(e\left(\frac{25}{52}\right)\) | \(e\left(\frac{7}{312}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{149}{312}\right)\) |
| \(\chi_{162240}(36359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{305}{312}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{293}{312}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{17}{312}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{139}{312}\right)\) |
| \(\chi_{162240}(37559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{311}{312}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{131}{312}\right)\) | \(e\left(\frac{49}{52}\right)\) | \(e\left(\frac{47}{312}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{109}{312}\right)\) |
| \(\chi_{162240}(38279,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{145}{312}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{37}{312}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{49}{312}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{107}{312}\right)\) |
| \(\chi_{162240}(41879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{235}{312}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{103}{312}\right)\) | \(e\left(\frac{29}{52}\right)\) | \(e\left(\frac{187}{312}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{281}{312}\right)\) |
| \(\chi_{162240}(42599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{197}{312}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{89}{312}\right)\) | \(e\left(\frac{19}{52}\right)\) | \(e\left(\frac{101}{312}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{55}{312}\right)\) |
| \(\chi_{162240}(43799,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{107}{312}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{23}{312}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{275}{312}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{193}{312}\right)\) |
| \(\chi_{162240}(44519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{109}{312}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{73}{312}\right)\) | \(e\left(\frac{15}{52}\right)\) | \(e\left(\frac{181}{312}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{287}{312}\right)\) |
| \(\chi_{162240}(48119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{271}{312}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{67}{312}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{55}{312}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{101}{312}\right)\) |
| \(\chi_{162240}(48839,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{89}{312}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{197}{312}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(e\left(\frac{185}{312}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{283}{312}\right)\) |
| \(\chi_{162240}(50039,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{215}{312}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{227}{312}\right)\) | \(e\left(\frac{21}{52}\right)\) | \(e\left(\frac{191}{312}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{277}{312}\right)\) |
| \(\chi_{162240}(50759,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{73}{312}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{109}{312}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{1}{312}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{155}{312}\right)\) |
| \(\chi_{162240}(54359,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{307}{312}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{31}{312}\right)\) | \(e\left(\frac{37}{52}\right)\) | \(e\left(\frac{235}{312}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{233}{312}\right)\) |