from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1288, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,0,5]))
pari: [g,chi] = znchar(Mod(43,1288))
Basic properties
| Modulus: | \(1288\) | |
| Conductor: | \(184\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{184}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1288.bj
\(\chi_{1288}(43,\cdot)\) \(\chi_{1288}(99,\cdot)\) \(\chi_{1288}(155,\cdot)\) \(\chi_{1288}(267,\cdot)\) \(\chi_{1288}(379,\cdot)\) \(\chi_{1288}(435,\cdot)\) \(\chi_{1288}(603,\cdot)\) \(\chi_{1288}(659,\cdot)\) \(\chi_{1288}(939,\cdot)\) \(\chi_{1288}(1275,\cdot)\)
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: | 22.22.339058325839400057321133061640411938816.1 |
Values on generators
\((967,645,185,281)\) → \((-1,-1,1,e\left(\frac{5}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 1288 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) |
sage: chi.jacobi_sum(n)