![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4729, base_ring=CyclotomicField(394))
M = H._module
chi = DirichletCharacter(H, M([81]))
![Copy content]()
gp:[g,chi] = znchar(Mod(96, 4729))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4729.96");
| Modulus: | \(4729\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(4729\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(394\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
gp:charorder(g,chi)
![Copy content]()
magma:Order(chi);
|
| Real: | no |
![Copy content]()
sage:chi.multiplicative_order() <= 2
![Copy content]()
gp:charorder(g,chi) <= 2
![Copy content]()
magma:Order(chi) le 2;
|
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{4729}(4,\cdot)\)
\(\chi_{4729}(6,\cdot)\)
\(\chi_{4729}(9,\cdot)\)
\(\chi_{4729}(64,\cdot)\)
\(\chi_{4729}(96,\cdot)\)
\(\chi_{4729}(144,\cdot)\)
\(\chi_{4729}(197,\cdot)\)
\(\chi_{4729}(216,\cdot)\)
\(\chi_{4729}(242,\cdot)\)
\(\chi_{4729}(254,\cdot)\)
\(\chi_{4729}(324,\cdot)\)
\(\chi_{4729}(350,\cdot)\)
\(\chi_{4729}(355,\cdot)\)
\(\chi_{4729}(363,\cdot)\)
\(\chi_{4729}(381,\cdot)\)
\(\chi_{4729}(413,\cdot)\)
\(\chi_{4729}(422,\cdot)\)
\(\chi_{4729}(451,\cdot)\)
\(\chi_{4729}(454,\cdot)\)
\(\chi_{4729}(455,\cdot)\)
\(\chi_{4729}(458,\cdot)\)
\(\chi_{4729}(475,\cdot)\)
\(\chi_{4729}(486,\cdot)\)
\(\chi_{4729}(525,\cdot)\)
\(\chi_{4729}(607,\cdot)\)
\(\chi_{4729}(613,\cdot)\)
\(\chi_{4729}(633,\cdot)\)
\(\chi_{4729}(634,\cdot)\)
\(\chi_{4729}(643,\cdot)\)
\(\chi_{4729}(670,\cdot)\)
...
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
![Copy content]()
magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\(17\) → \(e\left(\frac{81}{394}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4729 }(96, a) \) |
\(1\) | \(1\) | \(e\left(\frac{74}{197}\right)\) | \(e\left(\frac{187}{197}\right)\) | \(e\left(\frac{148}{197}\right)\) | \(e\left(\frac{185}{197}\right)\) | \(e\left(\frac{64}{197}\right)\) | \(e\left(\frac{5}{197}\right)\) | \(e\left(\frac{25}{197}\right)\) | \(e\left(\frac{177}{197}\right)\) | \(e\left(\frac{62}{197}\right)\) | \(e\left(\frac{55}{394}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
