from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,57]))
pari: [g,chi] = znchar(Mod(747,847))
Basic properties
| Modulus: | \(847\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 847.x
\(\chi_{847}(10,\cdot)\) \(\chi_{847}(54,\cdot)\) \(\chi_{847}(87,\cdot)\) \(\chi_{847}(131,\cdot)\) \(\chi_{847}(164,\cdot)\) \(\chi_{847}(208,\cdot)\) \(\chi_{847}(285,\cdot)\) \(\chi_{847}(318,\cdot)\) \(\chi_{847}(395,\cdot)\) \(\chi_{847}(439,\cdot)\) \(\chi_{847}(472,\cdot)\) \(\chi_{847}(516,\cdot)\) \(\chi_{847}(549,\cdot)\) \(\chi_{847}(593,\cdot)\) \(\chi_{847}(626,\cdot)\) \(\chi_{847}(670,\cdot)\) \(\chi_{847}(703,\cdot)\) \(\chi_{847}(747,\cdot)\) \(\chi_{847}(780,\cdot)\) \(\chi_{847}(824,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((122,365)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(747, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) |
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)