![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,22,40]))
![Copy content]()
pari:[g,chi] = znchar(Mod(748,805))
| Modulus: | \(805\) | |
| Conductor: | \(805\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(44\) |
![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)
|
\(\chi_{805}(13,\cdot)\)
\(\chi_{805}(27,\cdot)\)
\(\chi_{805}(48,\cdot)\)
\(\chi_{805}(62,\cdot)\)
\(\chi_{805}(118,\cdot)\)
\(\chi_{805}(167,\cdot)\)
\(\chi_{805}(188,\cdot)\)
\(\chi_{805}(202,\cdot)\)
\(\chi_{805}(223,\cdot)\)
\(\chi_{805}(307,\cdot)\)
\(\chi_{805}(328,\cdot)\)
\(\chi_{805}(363,\cdot)\)
\(\chi_{805}(377,\cdot)\)
\(\chi_{805}(468,\cdot)\)
\(\chi_{805}(538,\cdot)\)
\(\chi_{805}(587,\cdot)\)
\(\chi_{805}(657,\cdot)\)
\(\chi_{805}(692,\cdot)\)
\(\chi_{805}(748,\cdot)\)
\(\chi_{805}(762,\cdot)\)
![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 ]
\((162,346,281)\) → \((-i,-1,e\left(\frac{10}{11}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 805 }(748, a) \) |
\(1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) |
![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)
