from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3640, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,6,6,10,3]))
pari: [g,chi] = znchar(Mod(1139,3640))
Basic properties
Modulus: | \(3640\) | |
Conductor: | \(3640\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | 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 3640.iy
\(\chi_{3640}(619,\cdot)\) \(\chi_{3640}(1139,\cdot)\) \(\chi_{3640}(2579,\cdot)\) \(\chi_{3640}(3099,\cdot)\)
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
\((911,1821,1457,521,561)\) → \((-1,-1,-1,e\left(\frac{5}{6}\right),i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(33\) |
\( \chi_{ 3640 }(1139, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)