from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1587, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,7]))
pari: [g,chi] = znchar(Mod(1121,1587))
Basic properties
| Modulus: | \(1587\) | |
| Conductor: | \(69\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{69}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1587.g
\(\chi_{1587}(263,\cdot)\) \(\chi_{1587}(359,\cdot)\) \(\chi_{1587}(557,\cdot)\) \(\chi_{1587}(659,\cdot)\) \(\chi_{1587}(803,\cdot)\) \(\chi_{1587}(881,\cdot)\) \(\chi_{1587}(1100,\cdot)\) \(\chi_{1587}(1121,\cdot)\) \(\chi_{1587}(1253,\cdot)\) \(\chi_{1587}(1469,\cdot)\)
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: | \(\Q(\zeta_{69})^+\) |
Values on generators
\((530,1063)\) → \((-1,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 1587 }(1121, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)