from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1045, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,12,20]))
chi.galois_orbit()
[g,chi] = znchar(Mod(49,1045))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1045\) | |
| Conductor: | \(1045\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | 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: | \(\Q(\zeta_{15})\) |
| Fixed field: | Number field defined by a degree 30 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1045}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(-1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{1045}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(-1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{1045}(159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(-1\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{1045}(334,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(-1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{1045}(444,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(-1\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{1045}(619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(-1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{1045}(729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(-1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{1045}(999,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(-1\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) |