from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1127, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,27]))
pari: [g,chi] = znchar(Mod(471,1127))
Basic properties
| Modulus: | \(1127\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(149,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1127.x
\(\chi_{1127}(30,\cdot)\) \(\chi_{1127}(67,\cdot)\) \(\chi_{1127}(79,\cdot)\) \(\chi_{1127}(214,\cdot)\) \(\chi_{1127}(226,\cdot)\) \(\chi_{1127}(263,\cdot)\) \(\chi_{1127}(373,\cdot)\) \(\chi_{1127}(410,\cdot)\) \(\chi_{1127}(471,\cdot)\) \(\chi_{1127}(520,\cdot)\) \(\chi_{1127}(557,\cdot)\) \(\chi_{1127}(569,\cdot)\) \(\chi_{1127}(618,\cdot)\) \(\chi_{1127}(655,\cdot)\) \(\chi_{1127}(704,\cdot)\) \(\chi_{1127}(753,\cdot)\) \(\chi_{1127}(802,\cdot)\) \(\chi_{1127}(912,\cdot)\) \(\chi_{1127}(1010,\cdot)\) \(\chi_{1127}(1096,\cdot)\)
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
\((346,442)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 1127 }(471, a) \) | \(-1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) |
sage: chi.jacobi_sum(n)