sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(119, base_ring=CyclotomicField(4))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([2,1]))
pari: [g,chi] = znchar(Mod(13,119))
Basic properties
| Modulus: | \(119\) | |
| Conductor: | \(119\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 119.f
\(\chi_{119}(13,\cdot)\) \(\chi_{119}(55,\cdot)\)
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: | \(\Q(\sqrt{-1}) \) |
| Fixed field: | 4.0.240737.1 |
Values on generators
\((52,71)\) → \((-1,i)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \(-1\) | \(1\) | \(-1\) | \(-i\) | \(1\) | \(-i\) | \(i\) | \(-1\) | \(-1\) | \(i\) | \(-i\) | \(-i\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{119}(13,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(13,r) e\left(\frac{2r}{119}\right) = 6.7133546242+8.5983062105i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{119}(13,\cdot),\chi_{119}(1,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(13,r) \chi_{119}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{119}(13,·))
= \sum_{r \in \Z/119\Z}
\chi_{119}(13,r) e\left(\frac{1 r + 2 r^{-1}}{119}\right)
= 4.0445183743 \)