from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8007, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,3,14]))
pari: [g,chi] = znchar(Mod(1862,8007))
Basic properties
| Modulus: | \(8007\) | |
| Conductor: | \(8007\) | 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 8007.cg
\(\chi_{8007}(1862,\cdot)\) \(\chi_{8007}(1991,\cdot)\) \(\chi_{8007}(2405,\cdot)\) \(\chi_{8007}(4418,\cdot)\) \(\chi_{8007}(5159,\cdot)\) \(\chi_{8007}(7115,\cdot)\) \(\chi_{8007}(7172,\cdot)\) \(\chi_{8007}(7244,\cdot)\)
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
\((5339,1414,7855)\) → \((-1,e\left(\frac{1}{8}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 8007 }(1862, a) \) | \(1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)