from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,11,3]))
pari: [g,chi] = znchar(Mod(125,1288))
Basic properties
| Modulus: | \(1288\) | |
| Conductor: | \(1288\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1288.bk
\(\chi_{1288}(125,\cdot)\) \(\chi_{1288}(181,\cdot)\) \(\chi_{1288}(237,\cdot)\) \(\chi_{1288}(293,\cdot)\) \(\chi_{1288}(405,\cdot)\) \(\chi_{1288}(517,\cdot)\) \(\chi_{1288}(573,\cdot)\) \(\chi_{1288}(741,\cdot)\) \(\chi_{1288}(797,\cdot)\) \(\chi_{1288}(1077,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((967,645,185,281)\) → \((1,-1,-1,e\left(\frac{3}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 1288 }(125, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)