sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(26, base_ring=CyclotomicField(4))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([3]))
pari: [g,chi] = znchar(Mod(5,26))
Basic properties
| Modulus: | \(26\) | |
| Conductor: | \(13\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{13}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 26.d
\(\chi_{26}(5,\cdot)\) \(\chi_{26}(21,\cdot)\)
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
\(15\) → \(-i\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \(-1\) | \(1\) | \(1\) | \(-i\) | \(i\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(-i\) | \(i\) | \(-1\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-1}) \) |
| Fixed field: | 4.0.2197.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{26}(5,\cdot)) = \sum_{r\in \Z/26\Z} \chi_{26}(5,r) e\left(\frac{r}{13}\right) = 3.4508443768+1.0448316069i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{26}(5,\cdot),\chi_{26}(1,\cdot)) = \sum_{r\in \Z/26\Z} \chi_{26}(5,r) \chi_{26}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{26}(5,·))
= \sum_{r \in \Z/26\Z}
\chi_{26}(5,r) e\left(\frac{1 r + 2 r^{-1}}{26}\right)
= -1.473233513+-1.473233513i \)