from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([10]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,98))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(98\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(7\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 49.e | 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: | \(\Q(\zeta_{7})\) |
| Fixed field: | 7.7.13841287201.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{98}(15,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{98}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{98}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{98}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{98}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(1\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{98}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) |