from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4009, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([77,51]))
chi.galois_orbit()
[g,chi] = znchar(Mod(110,4009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4009\) | |
| Conductor: | \(4009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(126\) | 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: | $\Q(\zeta_{63})$ |
| Fixed field: | Number field defined by a degree 126 polynomial (not computed) |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4009}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{4009}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{4009}(516,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{4009}(659,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{4009}(801,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{4009}(870,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{4009}(876,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{4009}(1001,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{4009}(1021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{4009}(1212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{4009}(1515,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{4009}(1587,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{4009}(1826,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{4009}(1865,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{4009}(1932,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{4009}(2141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{4009}(2142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{4009}(2143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{4009}(2160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{4009}(2347,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{4009}(2352,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{4009}(2359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{4009}(2371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{4009}(2415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{4009}(2689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{4009}(2700,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{4009}(2986,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{4009}(3092,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{4009}(3620,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{4009}(3681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{4009}(3764,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{20}{21}\right)\) |