sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(96, base_ring=CyclotomicField(8))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,3,0]))
pari: [g,chi] = znchar(Mod(61,96))
Basic properties
| Modulus: | \(96\) | |
| Conductor: | \(32\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(8\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{32}(29,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 96.n
\(\chi_{96}(13,\cdot)\) \(\chi_{96}(37,\cdot)\) \(\chi_{96}(61,\cdot)\) \(\chi_{96}(85,\cdot)\)
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: | \(\Q(\zeta_{32})^+\) |
Values on generators
\((31,37,65)\) → \((1,e\left(\frac{3}{8}\right),1)\)
Values
| \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(-i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(-1\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{1}{8}\right)\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{96}(61,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(61,r) e\left(\frac{r}{48}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{96}(61,\cdot),\chi_{96}(1,\cdot)) = \sum_{r\in \Z/96\Z} \chi_{96}(61,r) \chi_{96}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{96}(61,·))
= \sum_{r \in \Z/96\Z}
\chi_{96}(61,r) e\left(\frac{1 r + 2 r^{-1}}{96}\right)
= 6.2855596671+-9.4070048194i \)