from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1856, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,8]))
pari: [g,chi] = znchar(Mod(111,1856))
Basic properties
| Modulus: | \(1856\) | |
| Conductor: | \(464\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{464}(227,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1856.bx
\(\chi_{1856}(111,\cdot)\) \(\chi_{1856}(239,\cdot)\) \(\chi_{1856}(335,\cdot)\) \(\chi_{1856}(431,\cdot)\) \(\chi_{1856}(687,\cdot)\) \(\chi_{1856}(719,\cdot)\) \(\chi_{1856}(1039,\cdot)\) \(\chi_{1856}(1167,\cdot)\) \(\chi_{1856}(1263,\cdot)\) \(\chi_{1856}(1359,\cdot)\) \(\chi_{1856}(1615,\cdot)\) \(\chi_{1856}(1647,\cdot)\)
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: | Number field defined by a degree 28 polynomial |
Values on generators
\((639,581,321)\) → \((-1,-i,e\left(\frac{2}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 1856 }(111, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{17}{28}\right)\) |
sage: chi.jacobi_sum(n)