sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([10,42]))
pari:[g,chi] = znchar(Mod(141,539))
Modulus: | \(539\) | |
Conductor: | \(539\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(35\) |
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)
|
\(\chi_{539}(15,\cdot)\)
\(\chi_{539}(36,\cdot)\)
\(\chi_{539}(64,\cdot)\)
\(\chi_{539}(71,\cdot)\)
\(\chi_{539}(92,\cdot)\)
\(\chi_{539}(113,\cdot)\)
\(\chi_{539}(141,\cdot)\)
\(\chi_{539}(169,\cdot)\)
\(\chi_{539}(190,\cdot)\)
\(\chi_{539}(218,\cdot)\)
\(\chi_{539}(225,\cdot)\)
\(\chi_{539}(267,\cdot)\)
\(\chi_{539}(302,\cdot)\)
\(\chi_{539}(323,\cdot)\)
\(\chi_{539}(372,\cdot)\)
\(\chi_{539}(379,\cdot)\)
\(\chi_{539}(400,\cdot)\)
\(\chi_{539}(421,\cdot)\)
\(\chi_{539}(449,\cdot)\)
\(\chi_{539}(456,\cdot)\)
\(\chi_{539}(477,\cdot)\)
\(\chi_{539}(498,\cdot)\)
\(\chi_{539}(526,\cdot)\)
\(\chi_{539}(533,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((199,442)\) → \((e\left(\frac{1}{7}\right),e\left(\frac{3}{5}\right))\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
\( \chi_{ 539 }(141, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) |
sage:chi.jacobi_sum(n)
sage:chi.gauss_sum(a)
pari:znchargauss(g,chi,a)
sage:chi.jacobi_sum(n)
sage:chi.kloosterman_sum(a,b)