from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8752, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([0,39,152]))
chi.galois_orbit()
[g,chi] = znchar(Mod(21,8752))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8752\) | |
| Conductor: | \(8752\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(156\) | 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: | $\Q(\zeta_{156})$ |
| Fixed field: | Number field defined by a degree 156 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8752}(21,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(-1\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{95}{156}\right)\) |
| \(\chi_{8752}(181,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(-1\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{31}{156}\right)\) |
| \(\chi_{8752}(325,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(-1\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{5}{156}\right)\) | \(e\left(\frac{115}{156}\right)\) |
| \(\chi_{8752}(445,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(-1\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{133}{156}\right)\) |
| \(\chi_{8752}(949,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(-1\) | \(e\left(\frac{151}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{79}{156}\right)\) |
| \(\chi_{8752}(1141,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{71}{156}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(-1\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{67}{156}\right)\) |
| \(\chi_{8752}(1293,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{125}{156}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(-1\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{119}{156}\right)\) | \(e\left(\frac{85}{156}\right)\) |
| \(\chi_{8752}(1325,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(-1\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{49}{156}\right)\) |
| \(\chi_{8752}(1333,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(-1\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{119}{156}\right)\) |
| \(\chi_{8752}(2309,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(-1\) | \(e\left(\frac{31}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{103}{156}\right)\) |
| \(\chi_{8752}(2317,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{145}{156}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(-1\) | \(e\left(\frac{53}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{5}{156}\right)\) |
| \(\chi_{8752}(2405,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(-1\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{131}{156}\right)\) |
| \(\chi_{8752}(2421,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(-1\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{29}{39}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{43}{156}\right)\) |
| \(\chi_{8752}(2629,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{11}{156}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(-1\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{151}{156}\right)\) |
| \(\chi_{8752}(2709,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{131}{156}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(-1\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{34}{39}\right)\) | \(e\left(\frac{101}{156}\right)\) | \(e\left(\frac{139}{156}\right)\) |
| \(\chi_{8752}(2789,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(-1\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{10}{39}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{35}{156}\right)\) |
| \(\chi_{8752}(3037,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(-1\) | \(e\left(\frac{17}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{28}{39}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{67}{156}\right)\) | \(e\left(\frac{137}{156}\right)\) |
| \(\chi_{8752}(3293,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(-1\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{14}{39}\right)\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{149}{156}\right)\) |
| \(\chi_{8752}(3701,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{7}{156}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(-1\) | \(e\left(\frac{23}{156}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{39}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{109}{156}\right)\) | \(e\left(\frac{11}{156}\right)\) |
| \(\chi_{8752}(3925,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(-1\) | \(e\left(\frac{139}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{39}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{19}{156}\right)\) |
| \(\chi_{8752}(3965,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(-1\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{19}{39}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{19}{156}\right)\) | \(e\left(\frac{125}{156}\right)\) |
| \(\chi_{8752}(4125,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(-1\) | \(e\left(\frac{89}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{103}{156}\right)\) | \(e\left(\frac{29}{156}\right)\) |
| \(\chi_{8752}(4293,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{127}{156}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(-1\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{61}{156}\right)\) | \(e\left(\frac{155}{156}\right)\) |
| \(\chi_{8752}(4317,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{121}{156}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(-1\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{39}\right)\) | \(e\left(\frac{32}{39}\right)\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{101}{156}\right)\) |
| \(\chi_{8752}(4397,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(-1\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{17}{156}\right)\) |
| \(\chi_{8752}(4557,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(-1\) | \(e\left(\frac{1}{156}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{20}{39}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{109}{156}\right)\) |
| \(\chi_{8752}(4701,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{137}{156}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(-1\) | \(e\left(\frac{49}{156}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{39}\right)\) | \(e\left(\frac{4}{39}\right)\) | \(e\left(\frac{83}{156}\right)\) | \(e\left(\frac{37}{156}\right)\) |
| \(\chi_{8752}(4821,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{35}{156}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(-1\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{38}{39}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{77}{156}\right)\) | \(e\left(\frac{55}{156}\right)\) |
| \(\chi_{8752}(5325,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{29}{156}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(-1\) | \(e\left(\frac{73}{156}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{17}{39}\right)\) | \(e\left(\frac{37}{39}\right)\) | \(e\left(\frac{95}{156}\right)\) | \(e\left(\frac{1}{156}\right)\) |
| \(\chi_{8752}(5517,\cdot)\) | \(1\) | \(1\) | \(i\) | \(e\left(\frac{149}{156}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(-1\) | \(e\left(\frac{133}{156}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{8}{39}\right)\) | \(e\left(\frac{22}{39}\right)\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{145}{156}\right)\) |
| \(\chi_{8752}(5669,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{47}{156}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(-1\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{39}\right)\) | \(e\left(\frac{25}{39}\right)\) | \(e\left(\frac{41}{156}\right)\) | \(e\left(\frac{7}{156}\right)\) |