from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4225, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([9,1]))
pari: [g,chi] = znchar(Mod(418,4225))
Basic properties
| Modulus: | \(4225\) | |
| Conductor: | \(65\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{65}(28,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4225.x
\(\chi_{4225}(418,\cdot)\) \(\chi_{4225}(657,\cdot)\) \(\chi_{4225}(3568,\cdot)\) \(\chi_{4225}(3807,\cdot)\)
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: | 12.12.3500313269603515625.1 |
Values on generators
\((677,3551)\) → \((-i,e\left(\frac{1}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 4225 }(418, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(-1\) |
sage: chi.jacobi_sum(n)