sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(294, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,1]))
pari: [g,chi] = znchar(Mod(31,294))
Basic properties
| Modulus: | \(294\) | |
| Conductor: | \(7\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{7}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 294.g
\(\chi_{294}(19,\cdot)\) \(\chi_{294}(31,\cdot)\)
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: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | \(\Q(\zeta_{7})\) |
Values on generators
\((197,199)\) → \((1,e\left(\frac{1}{6}\right))\)
Values
| \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(-1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{294}(31,\cdot)) = \sum_{r\in \Z/294\Z} \chi_{294}(31,r) e\left(\frac{r}{147}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{294}(31,\cdot),\chi_{294}(1,\cdot)) = \sum_{r\in \Z/294\Z} \chi_{294}(31,r) \chi_{294}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{294}(31,·))
= \sum_{r \in \Z/294\Z}
\chi_{294}(31,r) e\left(\frac{1 r + 2 r^{-1}}{294}\right)
= -24.1366607499+13.9353075813i \)