from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1568, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,18]))
pari: [g,chi] = znchar(Mod(55,1568))
Basic properties
| Modulus: | \(1568\) | |
| Conductor: | \(784\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{784}(643,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1568.bs
\(\chi_{1568}(55,\cdot)\) \(\chi_{1568}(167,\cdot)\) \(\chi_{1568}(279,\cdot)\) \(\chi_{1568}(503,\cdot)\) \(\chi_{1568}(615,\cdot)\) \(\chi_{1568}(727,\cdot)\) \(\chi_{1568}(839,\cdot)\) \(\chi_{1568}(951,\cdot)\) \(\chi_{1568}(1063,\cdot)\) \(\chi_{1568}(1287,\cdot)\) \(\chi_{1568}(1399,\cdot)\) \(\chi_{1568}(1511,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{28})\) |
| Fixed field: | 28.28.271776353216347717810469630450516372938858574109997048774397001728.1 |
Values on generators
\((1471,197,1473)\) → \((-1,-i,e\left(\frac{9}{14}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 1568 }(55, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(i\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) |
sage: chi.jacobi_sum(n)