![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(243, base_ring=CyclotomicField(162))
M = H._module
chi = DirichletCharacter(H, M([76]))
![Copy content]()
pari:[g,chi] = znchar(Mod(205,243))
| Modulus: | \(243\) | |
| Conductor: | \(243\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(81\) |
![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: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{243}(4,\cdot)\)
\(\chi_{243}(7,\cdot)\)
\(\chi_{243}(13,\cdot)\)
\(\chi_{243}(16,\cdot)\)
\(\chi_{243}(22,\cdot)\)
\(\chi_{243}(25,\cdot)\)
\(\chi_{243}(31,\cdot)\)
\(\chi_{243}(34,\cdot)\)
\(\chi_{243}(40,\cdot)\)
\(\chi_{243}(43,\cdot)\)
\(\chi_{243}(49,\cdot)\)
\(\chi_{243}(52,\cdot)\)
\(\chi_{243}(58,\cdot)\)
\(\chi_{243}(61,\cdot)\)
\(\chi_{243}(67,\cdot)\)
\(\chi_{243}(70,\cdot)\)
\(\chi_{243}(76,\cdot)\)
\(\chi_{243}(79,\cdot)\)
\(\chi_{243}(85,\cdot)\)
\(\chi_{243}(88,\cdot)\)
\(\chi_{243}(94,\cdot)\)
\(\chi_{243}(97,\cdot)\)
\(\chi_{243}(103,\cdot)\)
\(\chi_{243}(106,\cdot)\)
\(\chi_{243}(112,\cdot)\)
\(\chi_{243}(115,\cdot)\)
\(\chi_{243}(121,\cdot)\)
\(\chi_{243}(124,\cdot)\)
\(\chi_{243}(130,\cdot)\)
\(\chi_{243}(133,\cdot)\)
...
![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 ]
\(2\) → \(e\left(\frac{38}{81}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 243 }(205, a) \) |
\(1\) | \(1\) | \(e\left(\frac{38}{81}\right)\) | \(e\left(\frac{76}{81}\right)\) | \(e\left(\frac{64}{81}\right)\) | \(e\left(\frac{68}{81}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{62}{81}\right)\) | \(e\left(\frac{61}{81}\right)\) | \(e\left(\frac{25}{81}\right)\) | \(e\left(\frac{71}{81}\right)\) |
![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)
