from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(127, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([19]))
pari: [g,chi] = znchar(Mod(77,127))
Basic properties
Modulus: | \(127\) | |
Conductor: | \(127\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | 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 127.j
\(\chi_{127}(5,\cdot)\) \(\chi_{127}(10,\cdot)\) \(\chi_{127}(27,\cdot)\) \(\chi_{127}(33,\cdot)\) \(\chi_{127}(40,\cdot)\) \(\chi_{127}(51,\cdot)\) \(\chi_{127}(54,\cdot)\) \(\chi_{127}(66,\cdot)\) \(\chi_{127}(77,\cdot)\) \(\chi_{127}(80,\cdot)\) \(\chi_{127}(89,\cdot)\) \(\chi_{127}(102,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\(3\) → \(e\left(\frac{19}{42}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 127 }(77, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{16}{21}\right)\) |
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)