from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,3]))
pari: [g,chi] = znchar(Mod(32,847))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 847.w
\(\chi_{847}(32,\cdot)\) \(\chi_{847}(65,\cdot)\) \(\chi_{847}(109,\cdot)\) \(\chi_{847}(142,\cdot)\) \(\chi_{847}(186,\cdot)\) \(\chi_{847}(219,\cdot)\) \(\chi_{847}(263,\cdot)\) \(\chi_{847}(296,\cdot)\) \(\chi_{847}(340,\cdot)\) \(\chi_{847}(373,\cdot)\) \(\chi_{847}(417,\cdot)\) \(\chi_{847}(450,\cdot)\) \(\chi_{847}(494,\cdot)\) \(\chi_{847}(527,\cdot)\) \(\chi_{847}(571,\cdot)\) \(\chi_{847}(648,\cdot)\) \(\chi_{847}(681,\cdot)\) \(\chi_{847}(758,\cdot)\) \(\chi_{847}(802,\cdot)\) \(\chi_{847}(835,\cdot)\)
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
\((122,365)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(32, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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)