from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,8,30]))
pari: [g,chi] = znchar(Mod(691,3024))
Basic properties
| Modulus: | \(3024\) | |
| Conductor: | \(3024\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3024.hb
\(\chi_{3024}(187,\cdot)\) \(\chi_{3024}(283,\cdot)\) \(\chi_{3024}(691,\cdot)\) \(\chi_{3024}(787,\cdot)\) \(\chi_{3024}(1195,\cdot)\) \(\chi_{3024}(1291,\cdot)\) \(\chi_{3024}(1699,\cdot)\) \(\chi_{3024}(1795,\cdot)\) \(\chi_{3024}(2203,\cdot)\) \(\chi_{3024}(2299,\cdot)\) \(\chi_{3024}(2707,\cdot)\) \(\chi_{3024}(2803,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1135,757,785,2593)\) → \((-1,-i,e\left(\frac{2}{9}\right),e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3024 }(691, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)