from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3332, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([0,8,15]))
pari: [g,chi] = znchar(Mod(569,3332))
Basic properties
| Modulus: | \(3332\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{119}(93,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3332.bq
\(\chi_{3332}(569,\cdot)\) \(\chi_{3332}(961,\cdot)\) \(\chi_{3332}(1341,\cdot)\) \(\chi_{3332}(1549,\cdot)\) \(\chi_{3332}(1929,\cdot)\) \(\chi_{3332}(2321,\cdot)\) \(\chi_{3332}(2909,\cdot)\) \(\chi_{3332}(3313,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.2296127442650479958000916502307873630417.1 |
Values on generators
\((1667,885,785)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(23\) | \(25\) | \(27\) |
| \( \chi_{ 3332 }(569, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{7}{8}\right)\) |
sage: chi.jacobi_sum(n)