from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8040, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([6,6,6,9,2]))
pari: [g,chi] = znchar(Mod(1043,8040))
Basic properties
| Modulus: | \(8040\) | |
| Conductor: | \(8040\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8040.dk
\(\chi_{8040}(1043,\cdot)\) \(\chi_{8040}(4787,\cdot)\) \(\chi_{8040}(5867,\cdot)\) \(\chi_{8040}(8003,\cdot)\)
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
\((6031,4021,2681,3217,5161)\) → \((-1,-1,-1,-i,e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 8040 }(1043, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)