from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,20,6]))
pari: [g,chi] = znchar(Mod(619,3024))
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.gm
\(\chi_{3024}(115,\cdot)\) \(\chi_{3024}(355,\cdot)\) \(\chi_{3024}(619,\cdot)\) \(\chi_{3024}(859,\cdot)\) \(\chi_{3024}(1123,\cdot)\) \(\chi_{3024}(1363,\cdot)\) \(\chi_{3024}(1627,\cdot)\) \(\chi_{3024}(1867,\cdot)\) \(\chi_{3024}(2131,\cdot)\) \(\chi_{3024}(2371,\cdot)\) \(\chi_{3024}(2635,\cdot)\) \(\chi_{3024}(2875,\cdot)\)
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: | 36.36.13854191503209908618935635896029042342963612597539073353861947814992950774768801013176099229663232.2 |
Values on generators
\((1135,757,785,2593)\) → \((-1,i,e\left(\frac{5}{9}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 3024 }(619, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)