from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3751, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([63,22]))
pari: [g,chi] = znchar(Mod(87,3751))
Basic properties
| Modulus: | \(3751\) | |
| Conductor: | \(3751\) | 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 3751.cq
\(\chi_{3751}(87,\cdot)\) \(\chi_{3751}(98,\cdot)\) \(\chi_{3751}(428,\cdot)\) \(\chi_{3751}(439,\cdot)\) \(\chi_{3751}(769,\cdot)\) \(\chi_{3751}(780,\cdot)\) \(\chi_{3751}(1110,\cdot)\) \(\chi_{3751}(1121,\cdot)\) \(\chi_{3751}(1462,\cdot)\) \(\chi_{3751}(1792,\cdot)\) \(\chi_{3751}(1803,\cdot)\) \(\chi_{3751}(2133,\cdot)\) \(\chi_{3751}(2144,\cdot)\) \(\chi_{3751}(2474,\cdot)\) \(\chi_{3751}(2485,\cdot)\) \(\chi_{3751}(2815,\cdot)\) \(\chi_{3751}(2826,\cdot)\) \(\chi_{3751}(3156,\cdot)\) \(\chi_{3751}(3167,\cdot)\) \(\chi_{3751}(3497,\cdot)\)
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
\((2543,2421)\) → \((e\left(\frac{21}{22}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 3751 }(87, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) |
sage: chi.jacobi_sum(n)