from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,4]))
pari: [g,chi] = znchar(Mod(111,847))
Basic properties
| Modulus: | \(847\) | |
| Conductor: | \(847\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | 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.o
\(\chi_{847}(34,\cdot)\) \(\chi_{847}(111,\cdot)\) \(\chi_{847}(188,\cdot)\) \(\chi_{847}(265,\cdot)\) \(\chi_{847}(342,\cdot)\) \(\chi_{847}(419,\cdot)\) \(\chi_{847}(496,\cdot)\) \(\chi_{847}(573,\cdot)\) \(\chi_{847}(650,\cdot)\) \(\chi_{847}(804,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((122,365)\) → \((-1,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 847 }(111, a) \) | \(-1\) | \(1\) | \(e\left(\frac{2}{11}\right)\) | \(-1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(1\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) |
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)