sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(105, base_ring=CyclotomicField(2))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([1,0,0]))
pari: [g,chi] = znchar(Mod(71,105))
Basic properties
| Modulus: | \(105\) | |
| Conductor: | \(3\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | yes | |
| Primitive: | no, induced from \(\chi_{3}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 105.c
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: | \(\Q\) |
| Fixed field: | \(\Q(\sqrt{-3}) \) |
Values on generators
\((71,22,31)\) → \((-1,1,1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{105}(71,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(71,r) e\left(\frac{2r}{105}\right) = 1.7320508076i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{105}(71,\cdot),\chi_{105}(1,\cdot)) = \sum_{r\in \Z/105\Z} \chi_{105}(71,r) \chi_{105}(1,1-r) = -15 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{105}(71,·))
= \sum_{r \in \Z/105\Z}
\chi_{105}(71,r) e\left(\frac{1 r + 2 r^{-1}}{105}\right)
= 0.0 \)