sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(936, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,3,1,4]))
pari: [g,chi] = znchar(Mod(893,936))
Basic properties
| Modulus: | \(936\) | |
| Conductor: | \(936\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 936.cc
\(\chi_{936}(653,\cdot)\) \(\chi_{936}(893,\cdot)\)
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
\((703,469,209,145)\) → \((1,-1,e\left(\frac{1}{6}\right),e\left(\frac{2}{3}\right))\)
Values
| \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \(-1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | 6.0.287829075456.10 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{936}(893,\cdot)) = \sum_{r\in \Z/936\Z} \chi_{936}(893,r) e\left(\frac{r}{468}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{936}(893,\cdot),\chi_{936}(1,\cdot)) = \sum_{r\in \Z/936\Z} \chi_{936}(893,r) \chi_{936}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{936}(893,·))
= \sum_{r \in \Z/936\Z}
\chi_{936}(893,r) e\left(\frac{1 r + 2 r^{-1}}{936}\right)
= 1.1980823845+2.0751395616i \)