from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([40,28]))
pari: [g,chi] = znchar(Mod(256,637))
Basic properties
Modulus: | \(637\) | |
Conductor: | \(637\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 637.bi
\(\chi_{637}(16,\cdot)\) \(\chi_{637}(74,\cdot)\) \(\chi_{637}(107,\cdot)\) \(\chi_{637}(198,\cdot)\) \(\chi_{637}(256,\cdot)\) \(\chi_{637}(289,\cdot)\) \(\chi_{637}(347,\cdot)\) \(\chi_{637}(380,\cdot)\) \(\chi_{637}(438,\cdot)\) \(\chi_{637}(529,\cdot)\) \(\chi_{637}(562,\cdot)\) \(\chi_{637}(620,\cdot)\)
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 21 polynomial |
Values on generators
\((248,197)\) → \((e\left(\frac{20}{21}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 637 }(256, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) |
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)