from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,32,6]))
pari: [g,chi] = znchar(Mod(2005,3024))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3024.go
\(\chi_{3024}(229,\cdot)\) \(\chi_{3024}(493,\cdot)\) \(\chi_{3024}(733,\cdot)\) \(\chi_{3024}(997,\cdot)\) \(\chi_{3024}(1237,\cdot)\) \(\chi_{3024}(1501,\cdot)\) \(\chi_{3024}(1741,\cdot)\) \(\chi_{3024}(2005,\cdot)\) \(\chi_{3024}(2245,\cdot)\) \(\chi_{3024}(2509,\cdot)\) \(\chi_{3024}(2749,\cdot)\) \(\chi_{3024}(3013,\cdot)\)
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: | 36.0.13854191503209908618935635896029042342963612597539073353861947814992950774768801013176099229663232.1 |
Values on generators
\((1135,757,785,2593)\) → \((1,i,e\left(\frac{8}{9}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 3024 }(2005, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(-1\) | \(i\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)