from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7569, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([0,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(1108,7569))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(7569\) | |
| Conductor: | \(29\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 29.e | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{7})\) |
| Fixed field: | Number field defined by a degree 14 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{7569}(1108,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{7569}(1918,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{7569}(2719,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{7569}(4015,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{7569}(5950,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{7569}(6157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) |