![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(205, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,17]))
![Copy content]()
pari:[g,chi] = znchar(Mod(149,205))
| Modulus: | \(205\) | |
| Conductor: | \(205\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(40\) |
![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)
|
\(\chi_{205}(19,\cdot)\)
\(\chi_{205}(24,\cdot)\)
\(\chi_{205}(29,\cdot)\)
\(\chi_{205}(34,\cdot)\)
\(\chi_{205}(54,\cdot)\)
\(\chi_{205}(69,\cdot)\)
\(\chi_{205}(89,\cdot)\)
\(\chi_{205}(94,\cdot)\)
\(\chi_{205}(99,\cdot)\)
\(\chi_{205}(104,\cdot)\)
\(\chi_{205}(129,\cdot)\)
\(\chi_{205}(134,\cdot)\)
\(\chi_{205}(149,\cdot)\)
\(\chi_{205}(179,\cdot)\)
\(\chi_{205}(194,\cdot)\)
\(\chi_{205}(199,\cdot)\)
![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 ]
\((42,6)\) → \((-1,e\left(\frac{17}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 205 }(149, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(-i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
![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)
