sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(51, base_ring=CyclotomicField(8))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([4,7]))
pari: [g,chi] = znchar(Mod(2,51))
Basic properties
| Modulus: | \(51\) | |
| Conductor: | \(51\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | 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 51.g
\(\chi_{51}(2,\cdot)\) \(\chi_{51}(8,\cdot)\) \(\chi_{51}(26,\cdot)\) \(\chi_{51}(32,\cdot)\)
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: | \(\Q(\zeta_{8})\) |
| Fixed field: | 8.0.33237432513.1 |
Values on generators
\((35,37)\) → \((-1,e\left(\frac{7}{8}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \(-1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{3}{8}\right)\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{51}(2,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(2,r) e\left(\frac{2r}{51}\right) = 7.1336989848+-0.3321728377i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{51}(2,\cdot),\chi_{51}(1,\cdot)) = \sum_{r\in \Z/51\Z} \chi_{51}(2,r) \chi_{51}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{51}(2,·))
= \sum_{r \in \Z/51\Z}
\chi_{51}(2,r) e\left(\frac{1 r + 2 r^{-1}}{51}\right)
= 0.0 \)