from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,4,21]))
pari: [g,chi] = znchar(Mod(2923,3332))
Basic properties
| Modulus: | \(3332\) | |
| Conductor: | \(3332\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 3332.cc
\(\chi_{3332}(135,\cdot)\) \(\chi_{3332}(543,\cdot)\) \(\chi_{3332}(611,\cdot)\) \(\chi_{3332}(1019,\cdot)\) \(\chi_{3332}(1087,\cdot)\) \(\chi_{3332}(1495,\cdot)\) \(\chi_{3332}(1563,\cdot)\) \(\chi_{3332}(1971,\cdot)\) \(\chi_{3332}(2447,\cdot)\) \(\chi_{3332}(2515,\cdot)\) \(\chi_{3332}(2923,\cdot)\) \(\chi_{3332}(2991,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1667,885,785)\) → \((-1,e\left(\frac{2}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3332 }(2923, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)