from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7488, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,3,6,4]))
pari: [g,chi] = znchar(Mod(2447,7488))
Basic properties
| Modulus: | \(7488\) | |
| Conductor: | \(624\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{624}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7488.fs
\(\chi_{7488}(2447,\cdot)\) \(\chi_{7488}(3311,\cdot)\) \(\chi_{7488}(6191,\cdot)\) \(\chi_{7488}(7055,\cdot)\)
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: | 12.12.5108156609256675606528.1 |
Values on generators
\((703,6085,5825,5761)\) → \((-1,i,-1,e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 7488 }(2447, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)