from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,0,30,18,35]))
chi.galois_orbit()
[g,chi] = znchar(Mod(89,7800))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7800\) | |
| Conductor: | \(975\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 975.cw | 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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7800}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{7800}(929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(1289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{7800}(2009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(2489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(3209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{7800}(3569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(4409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{7800}(4769,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{7800}(5129,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(5609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(5969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{7800}(6329,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |
| \(\chi_{7800}(6689,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(7169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{3}\right)\) |
| \(\chi_{7800}(7529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{2}{3}\right)\) |