from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,63]))
pari: [g,chi] = znchar(Mod(934,1089))
Basic properties
Modulus: | \(1089\) | |
Conductor: | \(1089\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1089.x
\(\chi_{1089}(43,\cdot)\) \(\chi_{1089}(76,\cdot)\) \(\chi_{1089}(142,\cdot)\) \(\chi_{1089}(175,\cdot)\) \(\chi_{1089}(274,\cdot)\) \(\chi_{1089}(340,\cdot)\) \(\chi_{1089}(373,\cdot)\) \(\chi_{1089}(439,\cdot)\) \(\chi_{1089}(472,\cdot)\) \(\chi_{1089}(538,\cdot)\) \(\chi_{1089}(571,\cdot)\) \(\chi_{1089}(637,\cdot)\) \(\chi_{1089}(670,\cdot)\) \(\chi_{1089}(736,\cdot)\) \(\chi_{1089}(769,\cdot)\) \(\chi_{1089}(835,\cdot)\) \(\chi_{1089}(868,\cdot)\) \(\chi_{1089}(934,\cdot)\) \(\chi_{1089}(1033,\cdot)\) \(\chi_{1089}(1066,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((848,244)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
\( \chi_{ 1089 }(934, a) \) | \(-1\) | \(1\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)