from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6048, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,7,15]))
pari: [g,chi] = znchar(Mod(47,6048))
Basic properties
Modulus: | \(6048\) | |
Conductor: | \(1512\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1512}(803,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6048.fp
\(\chi_{6048}(47,\cdot)\) \(\chi_{6048}(815,\cdot)\) \(\chi_{6048}(2063,\cdot)\) \(\chi_{6048}(2831,\cdot)\) \(\chi_{6048}(4079,\cdot)\) \(\chi_{6048}(4847,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Fixed field: | 18.0.1882508498699710301766760487236393418883072.2 |
Values on generators
\((4159,3781,3809,2593)\) → \((-1,-1,e\left(\frac{7}{18}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 6048 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)