sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(28, base_ring=CyclotomicField(2))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([1,1]))
pari: [g,chi] = znchar(Mod(27,28))
Kronecker symbol representation
sage: kronecker_character(28)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{28}{\bullet}\right)\)
Basic properties
| Modulus: | \(28\) | |
| Conductor: | \(28\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 28.d
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
\((15,17)\) → \((-1,-1)\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) | \(1\) | \(-1\) | \(1\) |
Related number fields
| Field of values: | \(\Q\) |
| Fixed field: | \(\Q(\sqrt{7}) \) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{28}(27,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(27,r) e\left(\frac{r}{14}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{28}(27,\cdot),\chi_{28}(1,\cdot)) = \sum_{r\in \Z/28\Z} \chi_{28}(27,r) \chi_{28}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{28}(27,·))
= \sum_{r \in \Z/28\Z}
\chi_{28}(27,r) e\left(\frac{1 r + 2 r^{-1}}{28}\right)
= 2.354940211 \)