from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8008, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,0,40,48,25]))
chi.galois_orbit()
[g,chi] = znchar(Mod(487,8008))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8008\) | |
| Conductor: | \(4004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 4004.ip | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) | \(29\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8008}(487,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{8008}(1215,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{8008}(1367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{8008}(1775,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{8008}(2039,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{8008}(2095,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{8008}(2671,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{8008}(3551,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{8008}(4855,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{8008}(5415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{8008}(5679,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{8008}(5735,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{8008}(6143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{8008}(6407,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{8008}(7599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{8008}(7863,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) |