from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4029, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,20]))
pari: [g,chi] = znchar(Mod(3137,4029))
Basic properties
Modulus: | \(4029\) | |
Conductor: | \(4029\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4029.bi
\(\chi_{4029}(767,\cdot)\) \(\chi_{4029}(893,\cdot)\) \(\chi_{4029}(1130,\cdot)\) \(\chi_{4029}(1715,\cdot)\) \(\chi_{4029}(2552,\cdot)\) \(\chi_{4029}(3137,\cdot)\) \(\chi_{4029}(3374,\cdot)\) \(\chi_{4029}(3500,\cdot)\)
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
\((2687,3556,3163)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 4029 }(3137, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(-i\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)