sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(9, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(2,9))
Basic properties
| Modulus: | \(9\) | |
| Conductor: | \(9\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6\) | 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 9.d
\(\chi_{9}(2,\cdot)\) \(\chi_{9}(5,\cdot)\)
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
\(2\) → \(e\left(\frac{1}{6}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) |
| \(-1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | \(\Q(\zeta_{9})\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{9}(2,\cdot)) = \sum_{r\in \Z/9\Z} \chi_{9}(2,r) e\left(\frac{2r}{9}\right) = 0.520944533+2.954423259i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{9}(2,\cdot),\chi_{9}(1,\cdot)) = \sum_{r\in \Z/9\Z} \chi_{9}(2,r) \chi_{9}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{9}(2,·))
= \sum_{r \in \Z/9\Z}
\chi_{9}(2,r) e\left(\frac{1 r + 2 r^{-1}}{9}\right)
= -1.5+2.5980762114i \)