from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(199, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([5]))
pari: [g,chi] = znchar(Mod(12,199))
Basic properties
Modulus: | \(199\) | |
Conductor: | \(199\) | 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 199.j
\(\chi_{199}(12,\cdot)\) \(\chi_{199}(17,\cdot)\) \(\chi_{199}(27,\cdot)\) \(\chi_{199}(42,\cdot)\) \(\chi_{199}(55,\cdot)\) \(\chi_{199}(59,\cdot)\) \(\chi_{199}(67,\cdot)\) \(\chi_{199}(76,\cdot)\) \(\chi_{199}(82,\cdot)\) \(\chi_{199}(83,\cdot)\) \(\chi_{199}(88,\cdot)\) \(\chi_{199}(101,\cdot)\) \(\chi_{199}(109,\cdot)\) \(\chi_{199}(135,\cdot)\) \(\chi_{199}(147,\cdot)\) \(\chi_{199}(159,\cdot)\) \(\chi_{199}(171,\cdot)\) \(\chi_{199}(174,\cdot)\) \(\chi_{199}(191,\cdot)\) \(\chi_{199}(194,\cdot)\)
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
\(3\) → \(e\left(\frac{5}{66}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 199 }(12, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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)