![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,21,12]))
![Copy content]()
pari:[g,chi] = znchar(Mod(323,735))
| Modulus: | \(735\) | |
| Conductor: | \(735\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(28\) |
![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)
|
\(\chi_{735}(8,\cdot)\)
\(\chi_{735}(92,\cdot)\)
\(\chi_{735}(113,\cdot)\)
\(\chi_{735}(218,\cdot)\)
\(\chi_{735}(302,\cdot)\)
\(\chi_{735}(323,\cdot)\)
\(\chi_{735}(407,\cdot)\)
\(\chi_{735}(428,\cdot)\)
\(\chi_{735}(512,\cdot)\)
\(\chi_{735}(533,\cdot)\)
\(\chi_{735}(617,\cdot)\)
\(\chi_{735}(722,\cdot)\)
![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 ]
\((491,442,346)\) → \((-1,-i,e\left(\frac{3}{7}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 735 }(323, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(-1\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) |
![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)
