from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([5,21]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,2009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2009\) | |
| Conductor: | \(287\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 287.x | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{15})\) |
| Fixed field: | 30.0.47056013503315267757707869738647378732273238450224554604301383567.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2009}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{2009}(558,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{2009}(619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{2009}(1048,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{2009}(1097,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{2009}(1501,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{2009}(1685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{2009}(1991,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) |