from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3963, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,5]))
pari: [g,chi] = znchar(Mod(2267,3963))
Basic properties
| Modulus: | \(3963\) | |
| Conductor: | \(3963\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 3963.bi
\(\chi_{3963}(71,\cdot)\) \(\chi_{3963}(80,\cdot)\) \(\chi_{3963}(155,\cdot)\) \(\chi_{3963}(548,\cdot)\) \(\chi_{3963}(614,\cdot)\) \(\chi_{3963}(644,\cdot)\) \(\chi_{3963}(677,\cdot)\) \(\chi_{3963}(707,\cdot)\) \(\chi_{3963}(773,\cdot)\) \(\chi_{3963}(1166,\cdot)\) \(\chi_{3963}(1241,\cdot)\) \(\chi_{3963}(1250,\cdot)\) \(\chi_{3963}(1376,\cdot)\) \(\chi_{3963}(1685,\cdot)\) \(\chi_{3963}(2267,\cdot)\) \(\chi_{3963}(2618,\cdot)\) \(\chi_{3963}(2666,\cdot)\) \(\chi_{3963}(3017,\cdot)\) \(\chi_{3963}(3599,\cdot)\) \(\chi_{3963}(3908,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((1322,13)\) → \((-1,e\left(\frac{5}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 3963 }(2267, a) \) | \(-1\) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)