from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,1]))
pari: [g,chi] = znchar(Mod(591,784))
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 \(\chi_{196}(3,\cdot)\) | 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()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | \(\Q(\zeta_{196})^+\) |
Values on generators
\((687,197,689)\) → \((-1,1,e\left(\frac{1}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 784 }(591, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)