sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(28, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([3,2]))
pari: [g,chi] = znchar(Mod(23,28))
Basic properties
| Modulus: | \(28\) | |
| Conductor: | \(28\) | 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 28.g
\(\chi_{28}(11,\cdot)\) \(\chi_{28}(23,\cdot)\)
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
\((15,17)\) → \((-1,e\left(\frac{1}{3}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | 6.0.153664.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{28}(23,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(23,r) e\left(\frac{r}{14}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{28}(23,\cdot),\chi_{28}(1,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(23,r) \chi_{28}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{28}(23,·))
= \sum_{r \in \Z/28\Z}
\chi_{28}(23,r) e\left(\frac{1 r + 2 r^{-1}}{28}\right)
= 2.9306671564+1.6920214716i \)