sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(64, base_ring=CyclotomicField(4))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([2,1]))
pari: [g,chi] = znchar(Mod(15,64))
Basic properties
| Modulus: | \(64\) | |
| Conductor: | \(16\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{16}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 64.f
\(\chi_{64}(15,\cdot)\) \(\chi_{64}(47,\cdot)\)
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: | \(\Q(\sqrt{-1}) \) |
| Fixed field: | 4.0.2048.2 |
Values on generators
\((63,5)\) → \((-1,i)\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \(-1\) | \(1\) | \(i\) | \(i\) | \(1\) | \(-1\) | \(-i\) | \(-i\) | \(-1\) | \(1\) | \(i\) | \(i\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{64}(15,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(15,r) e\left(\frac{r}{32}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{64}(15,\cdot),\chi_{64}(1,\cdot)) = \sum_{r\in \Z/64\Z} \chi_{64}(15,r) \chi_{64}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{64}(15,·))
= \sum_{r \in \Z/64\Z}
\chi_{64}(15,r) e\left(\frac{1 r + 2 r^{-1}}{64}\right)
= 0.0 \)