from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,3]))
pari: [g,chi] = znchar(Mod(32,1089))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1089.y
\(\chi_{1089}(32,\cdot)\) \(\chi_{1089}(65,\cdot)\) \(\chi_{1089}(131,\cdot)\) \(\chi_{1089}(164,\cdot)\) \(\chi_{1089}(230,\cdot)\) \(\chi_{1089}(263,\cdot)\) \(\chi_{1089}(329,\cdot)\) \(\chi_{1089}(428,\cdot)\) \(\chi_{1089}(461,\cdot)\) \(\chi_{1089}(527,\cdot)\) \(\chi_{1089}(560,\cdot)\) \(\chi_{1089}(626,\cdot)\) \(\chi_{1089}(659,\cdot)\) \(\chi_{1089}(758,\cdot)\) \(\chi_{1089}(824,\cdot)\) \(\chi_{1089}(857,\cdot)\) \(\chi_{1089}(923,\cdot)\) \(\chi_{1089}(956,\cdot)\) \(\chi_{1089}(1022,\cdot)\) \(\chi_{1089}(1055,\cdot)\)
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
\((848,244)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1089 }(32, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) |
sage: chi.jacobi_sum(n)