sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(1323, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([10,12]))
pari: [g,chi] = znchar(Mod(214,1323))
Basic properties
| Modulus: | \(1323\) | |
| Conductor: | \(189\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{189}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1323.x
\(\chi_{1323}(214,\cdot)\) \(\chi_{1323}(373,\cdot)\) \(\chi_{1323}(655,\cdot)\) \(\chi_{1323}(814,\cdot)\) \(\chi_{1323}(1096,\cdot)\) \(\chi_{1323}(1255,\cdot)\)
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
\((785,1081)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{2}{3}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(1\) | \(1\) |
Related number fields
| Field of values: | \(\Q(\zeta_{9})\) |
| Fixed field: | 9.9.3691950281939241.2 |