sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(1339, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([2,2]))
pari: [g,chi] = znchar(Mod(159,1339))
Basic properties
| Modulus: | \(1339\) | |
| Conductor: | \(1339\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3\) | 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 1339.h
\(\chi_{1339}(159,\cdot)\) \(\chi_{1339}(1179,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\((1237,417)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{1}{3}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | 3.3.1792921.2 |