from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2856, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,6,6,10,9]))
pari: [g,chi] = znchar(Mod(1517,2856))
Basic properties
| Modulus: | \(2856\) | |
| Conductor: | \(2856\) | 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 2856.eg
\(\chi_{2856}(1109,\cdot)\) \(\chi_{2856}(1517,\cdot)\) \(\chi_{2856}(1781,\cdot)\) \(\chi_{2856}(2189,\cdot)\)
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: | 12.12.6401594233356076787154812928.1 |
Values on generators
\((2143,1429,953,409,2689)\) → \((1,-1,-1,e\left(\frac{5}{6}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2856 }(1517, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)