![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([39,76]))
![Copy content]()
pari:[g,chi] = znchar(Mod(42,845))
| Modulus: | \(845\) | |
| Conductor: | \(845\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(156\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{845}(3,\cdot)\)
\(\chi_{845}(42,\cdot)\)
\(\chi_{845}(48,\cdot)\)
\(\chi_{845}(68,\cdot)\)
\(\chi_{845}(87,\cdot)\)
\(\chi_{845}(107,\cdot)\)
\(\chi_{845}(113,\cdot)\)
\(\chi_{845}(133,\cdot)\)
\(\chi_{845}(152,\cdot)\)
\(\chi_{845}(172,\cdot)\)
\(\chi_{845}(178,\cdot)\)
\(\chi_{845}(198,\cdot)\)
\(\chi_{845}(217,\cdot)\)
\(\chi_{845}(237,\cdot)\)
\(\chi_{845}(243,\cdot)\)
\(\chi_{845}(263,\cdot)\)
\(\chi_{845}(282,\cdot)\)
\(\chi_{845}(302,\cdot)\)
\(\chi_{845}(308,\cdot)\)
\(\chi_{845}(328,\cdot)\)
\(\chi_{845}(347,\cdot)\)
\(\chi_{845}(367,\cdot)\)
\(\chi_{845}(373,\cdot)\)
\(\chi_{845}(393,\cdot)\)
\(\chi_{845}(412,\cdot)\)
\(\chi_{845}(432,\cdot)\)
\(\chi_{845}(438,\cdot)\)
\(\chi_{845}(458,\cdot)\)
\(\chi_{845}(477,\cdot)\)
\(\chi_{845}(497,\cdot)\)
...
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((677,171)\) → \((i,e\left(\frac{19}{39}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 845 }(42, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{115}{156}\right)\) | \(e\left(\frac{25}{156}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{35}{39}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{11}{52}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{7}{39}\right)\) | \(e\left(\frac{33}{52}\right)\) | \(e\left(\frac{3}{26}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
pari:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
