sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(3744, base_ring=CyclotomicField(12))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([6,6,2,11]))
pari: [g,chi] = znchar(Mod(1775,3744))
Basic properties
| Modulus: | \(3744\) | |
| Conductor: | \(936\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{936}(371,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3744.ga
\(\chi_{3744}(1103,\cdot)\) \(\chi_{3744}(1679,\cdot)\) \(\chi_{3744}(1775,\cdot)\) \(\chi_{3744}(3503,\cdot)\)
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
\((703,2341,2081,2017)\) → \((-1,-1,e\left(\frac{1}{6}\right),e\left(\frac{11}{12}\right))\)
Values
| \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \(-1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.0.182011731961249064312635392.2 |