from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,4,3]))
pari: [g,chi] = znchar(Mod(1696,6069))
Basic properties
| Modulus: | \(6069\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{119}(30,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6069.bb
\(\chi_{6069}(1696,\cdot)\) \(\chi_{6069}(3217,\cdot)\) \(\chi_{6069}(4951,\cdot)\) \(\chi_{6069}(6031,\cdot)\)
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: | 12.12.683635509017782097.1 |
Values on generators
\((2024,4336,3760)\) → \((1,e\left(\frac{1}{3}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(19\) | \(20\) |
| \( \chi_{ 6069 }(1696, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)