from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6040, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,0,9,4]))
pari: [g,chi] = znchar(Mod(183,6040))
Basic properties
Modulus: | \(6040\) | |
Conductor: | \(3020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3020}(183,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6040.cj
\(\chi_{6040}(183,\cdot)\) \(\chi_{6040}(2383,\cdot)\) \(\chi_{6040}(3807,\cdot)\) \(\chi_{6040}(6007,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
Fixed field: | Number field defined by a degree 12 polynomial |
Values on generators
\((1511,3021,2417,761)\) → \((-1,1,-i,e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 6040 }(183, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)