from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1089, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,6]))
pari: [g,chi] = znchar(Mod(1024,1089))
Basic properties
| Modulus: | \(1089\) | |
| Conductor: | \(1089\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | 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.u
\(\chi_{1089}(34,\cdot)\) \(\chi_{1089}(67,\cdot)\) \(\chi_{1089}(133,\cdot)\) \(\chi_{1089}(166,\cdot)\) \(\chi_{1089}(232,\cdot)\) \(\chi_{1089}(265,\cdot)\) \(\chi_{1089}(331,\cdot)\) \(\chi_{1089}(430,\cdot)\) \(\chi_{1089}(463,\cdot)\) \(\chi_{1089}(529,\cdot)\) \(\chi_{1089}(562,\cdot)\) \(\chi_{1089}(628,\cdot)\) \(\chi_{1089}(661,\cdot)\) \(\chi_{1089}(760,\cdot)\) \(\chi_{1089}(826,\cdot)\) \(\chi_{1089}(859,\cdot)\) \(\chi_{1089}(925,\cdot)\) \(\chi_{1089}(958,\cdot)\) \(\chi_{1089}(1024,\cdot)\) \(\chi_{1089}(1057,\cdot)\)
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 33 polynomial |
Values on generators
\((848,244)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1089 }(1024, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)