sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(930, base_ring=CyclotomicField(12))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([6,9,4]))
pari: [g,chi] = znchar(Mod(893,930))
Basic properties
| Modulus: | \(930\) | |
| Conductor: | \(465\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{465}(428,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 930.bf
\(\chi_{930}(377,\cdot)\) \(\chi_{930}(563,\cdot)\) \(\chi_{930}(707,\cdot)\) \(\chi_{930}(893,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{12})\) |
| Fixed field: | 12.12.1214370246668923828125.1 |
Values on generators
\((311,187,871)\) → \((-1,-i,e\left(\frac{1}{3}\right))\)
Values
| \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(-i\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{930}(893,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(893,r) e\left(\frac{r}{465}\right) = 3.2746816967+21.3137622157i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{930}(893,\cdot),\chi_{930}(1,\cdot)) = \sum_{r\in \Z/930\Z} \chi_{930}(893,r) \chi_{930}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{930}(893,·))
= \sum_{r \in \Z/930\Z}
\chi_{930}(893,r) e\left(\frac{1 r + 2 r^{-1}}{930}\right)
= -0.0 \)