sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([10,27]))
pari:[g,chi] = znchar(Mod(113,135))
Modulus: | \(135\) | |
Conductor: | \(135\) |
sage:chi.conductor()
pari:znconreyconductor(g,chi)
|
Order: | \(36\) |
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)
|
\(\chi_{135}(2,\cdot)\)
\(\chi_{135}(23,\cdot)\)
\(\chi_{135}(32,\cdot)\)
\(\chi_{135}(38,\cdot)\)
\(\chi_{135}(47,\cdot)\)
\(\chi_{135}(68,\cdot)\)
\(\chi_{135}(77,\cdot)\)
\(\chi_{135}(83,\cdot)\)
\(\chi_{135}(92,\cdot)\)
\(\chi_{135}(113,\cdot)\)
\(\chi_{135}(122,\cdot)\)
\(\chi_{135}(128,\cdot)\)
sage:chi.galois_orbit()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((56,82)\) → \((e\left(\frac{5}{18}\right),-i)\)
\(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 135 }(113, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) |
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)