from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(191, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([23]))
pari: [g,chi] = znchar(Mod(161,191))
Basic properties
Modulus: | \(191\) | |
Conductor: | \(191\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | 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 191.f
\(\chi_{191}(11,\cdot)\) \(\chi_{191}(14,\cdot)\) \(\chi_{191}(31,\cdot)\) \(\chi_{191}(37,\cdot)\) \(\chi_{191}(38,\cdot)\) \(\chi_{191}(41,\cdot)\) \(\chi_{191}(55,\cdot)\) \(\chi_{191}(66,\cdot)\) \(\chi_{191}(70,\cdot)\) \(\chi_{191}(84,\cdot)\) \(\chi_{191}(122,\cdot)\) \(\chi_{191}(139,\cdot)\) \(\chi_{191}(155,\cdot)\) \(\chi_{191}(159,\cdot)\) \(\chi_{191}(161,\cdot)\) \(\chi_{191}(166,\cdot)\) \(\chi_{191}(185,\cdot)\) \(\chi_{191}(186,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | Number field defined by a degree 38 polynomial |
Values on generators
\(19\) → \(e\left(\frac{23}{38}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 191 }(161, a) \) | \(-1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(-1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) |
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)