sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(10, base_ring=CyclotomicField(4))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([3]))
pari: [g,chi] = znchar(Mod(3,10))
Basic properties
| Modulus: | \(10\) | |
| Conductor: | \(5\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{5}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 10.c
\(\chi_{10}(3,\cdot)\) \(\chi_{10}(7,\cdot)\)
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
\(7\) → \(-i\)
Values
| \(-1\) | \(1\) | \(3\) | \(7\) |
| \(-1\) | \(1\) | \(i\) | \(-i\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-1}) \) |
| Fixed field: | \(\Q(\zeta_{5})\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{10}(3,\cdot)) = \sum_{r\in \Z/10\Z} \chi_{10}(3,r) e\left(\frac{r}{5}\right) = 1.1755705046+1.9021130326i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{10}(3,\cdot),\chi_{10}(1,\cdot)) = \sum_{r\in \Z/10\Z} \chi_{10}(3,r) \chi_{10}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{10}(3,·))
= \sum_{r \in \Z/10\Z}
\chi_{10}(3,r) e\left(\frac{1 r + 2 r^{-1}}{10}\right)
= 1.9021130326+1.9021130326i \)