from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,2,3]))
pari: [g,chi] = znchar(Mod(587,8619))
Basic properties
Modulus: | \(8619\) | |
Conductor: | \(663\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{663}(587,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8619.cg
\(\chi_{8619}(587,\cdot)\) \(\chi_{8619}(1103,\cdot)\) \(\chi_{8619}(1709,\cdot)\) \(\chi_{8619}(3629,\cdot)\) \(\chi_{8619}(4751,\cdot)\) \(\chi_{8619}(5051,\cdot)\) \(\chi_{8619}(6173,\cdot)\) \(\chi_{8619}(8600,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((5747,5917,2536)\) → \((-1,e\left(\frac{1}{12}\right),e\left(\frac{1}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
\( \chi_{ 8619 }(587, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)