sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(2960, base_ring=CyclotomicField(12))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([6,9,0,2]))
pari: [g,chi] = znchar(Mod(2691,2960))
Basic properties
| Modulus: | \(2960\) | |
| Conductor: | \(592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{592}(323,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2960.eb
\(\chi_{2960}(11,\cdot)\) \(\chi_{2960}(1211,\cdot)\) \(\chi_{2960}(1491,\cdot)\) \(\chi_{2960}(2691,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.0.41305425239182691803332608.1 |
Values on generators
\((2591,741,1777,2481)\) → \((-1,-i,1,e\left(\frac{1}{6}\right))\)
Values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2960 }(2691, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(-1\) | \(i\) |
sage: chi.jacobi_sum(n)