from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9898, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,11]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4311,9898))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(9898\) | |
| Conductor: | \(707\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 707.s | 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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.20.34126087659566522355396420380280630318342238349.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{9898}(4311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) |
| \(\chi_{9898}(4605,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{9898}(4703,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{9898}(5291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{9898}(7839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) |
| \(\chi_{9898}(8427,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) |
| \(\chi_{9898}(8525,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) |
| \(\chi_{9898}(8819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) |