sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(7, base_ring=CyclotomicField(2))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(6,7))
Kronecker symbol representation
sage: kronecker_character(-7)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{-7}{\bullet}\right)\)
Basic properties
| Modulus: | \(7\) | |
| Conductor: | \(7\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | yes | |
| 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 7.b
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\(3\) → \(-1\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
| \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
Related number fields
| Field of values: | \(\Q\) |
| Fixed field: | \(\Q(\sqrt{-7}) \) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{7}(6,\cdot)) = \sum_{r\in \Z/7\Z} \chi_{7}(6,r) e\left(\frac{2r}{7}\right) = 2.6457513111i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{7}(6,\cdot),\chi_{7}(1,\cdot)) = \sum_{r\in \Z/7\Z} \chi_{7}(6,r) \chi_{7}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{7}(6,·))
= \sum_{r \in \Z/7\Z}
\chi_{7}(6,r) e\left(\frac{1 r + 2 r^{-1}}{7}\right)
= 3.2991979214i \)