sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(8, base_ring=CyclotomicField(2))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,0]))
pari: [g,chi] = znchar(Mod(1,8))
Basic properties
| Modulus: | \(8\) | |
| Conductor: | \(1\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | yes | |
| Primitive: | no, induced from \(\chi_{1}(1,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8.a
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: | \(\Q\) |
| Fixed field: | \(\Q\) |
Values on generators
\((7,5)\) → \((1,1)\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) |
| \(1\) | \(1\) | \(1\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{8}(1,\cdot)) = \sum_{r\in \Z/8\Z} \chi_{8}(1,r) e\left(\frac{r}{4}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{8}(1,\cdot),\chi_{8}(1,\cdot)) = \sum_{r\in \Z/8\Z} \chi_{8}(1,r) \chi_{8}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{8}(1,·))
= \sum_{r \in \Z/8\Z}
\chi_{8}(1,r) e\left(\frac{1 r + 2 r^{-1}}{8}\right)
= 0.0 \)