sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(784, base_ring=CyclotomicField(42))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([21,0,5]))
pari: [g,chi] = znchar(Mod(47,784))
Basic properties
| Modulus: | \(784\) | |
| Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{196}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 784.bp
\(\chi_{784}(47,\cdot)\) \(\chi_{784}(143,\cdot)\) \(\chi_{784}(159,\cdot)\) \(\chi_{784}(255,\cdot)\) \(\chi_{784}(271,\cdot)\) \(\chi_{784}(367,\cdot)\) \(\chi_{784}(383,\cdot)\) \(\chi_{784}(479,\cdot)\) \(\chi_{784}(495,\cdot)\) \(\chi_{784}(591,\cdot)\) \(\chi_{784}(703,\cdot)\) \(\chi_{784}(719,\cdot)\)
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
\((687,197,689)\) → \((-1,1,e\left(\frac{5}{42}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | \(\Q(\zeta_{196})^+\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{784}(47,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(47,r) e\left(\frac{r}{392}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{784}(47,\cdot),\chi_{784}(1,\cdot)) = \sum_{r\in \Z/784\Z} \chi_{784}(47,r) \chi_{784}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{784}(47,·))
= \sum_{r \in \Z/784\Z}
\chi_{784}(47,r) e\left(\frac{1 r + 2 r^{-1}}{784}\right)
= 0.0 \)