from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3328, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,15,8]))
pari: [g,chi] = znchar(Mod(159,3328))
Basic properties
| Modulus: | \(3328\) | |
| Conductor: | \(416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{416}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3328.cg
\(\chi_{3328}(159,\cdot)\) \(\chi_{3328}(607,\cdot)\) \(\chi_{3328}(991,\cdot)\) \(\chi_{3328}(1439,\cdot)\) \(\chi_{3328}(1823,\cdot)\) \(\chi_{3328}(2271,\cdot)\) \(\chi_{3328}(2655,\cdot)\) \(\chi_{3328}(3103,\cdot)\)
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: | Number field defined by a degree 24 polynomial |
Values on generators
\((1535,261,769)\) → \((-1,e\left(\frac{5}{8}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3328 }(159, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)