sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(93, base_ring=CyclotomicField(2))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([1,1]))
pari: [g,chi] = znchar(Mod(92,93))
Kronecker symbol representation
sage: kronecker_character(93)
pari: znchartokronecker(g,chi)
\(\displaystyle\left(\frac{93}{\bullet}\right)\)
Basic properties
| Modulus: | \(93\) | |
| Conductor: | \(93\) | 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 93.c
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: | \(\Q(\sqrt{93}) \) |
Values on generators
\((32,34)\) → \((-1,-1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(-1\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{93}(92,\cdot)) = \sum_{r\in \Z/93\Z} \chi_{93}(92,r) e\left(\frac{2r}{93}\right) = -9.643650761 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{93}(92,\cdot),\chi_{93}(1,\cdot)) = \sum_{r\in \Z/93\Z} \chi_{93}(92,r) \chi_{93}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{93}(92,·))
= \sum_{r \in \Z/93\Z}
\chi_{93}(92,r) e\left(\frac{1 r + 2 r^{-1}}{93}\right)
= 0.0 \)