![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(835, base_ring=CyclotomicField(332))
M = H._module
chi = DirichletCharacter(H, M([249,228]))
![Copy content]()
gp:[g,chi] = znchar(Mod(128, 835))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("835.128");
| Modulus: | \(835\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(835\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(332\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{835}(2,\cdot)\)
\(\chi_{835}(3,\cdot)\)
\(\chi_{835}(7,\cdot)\)
\(\chi_{835}(8,\cdot)\)
\(\chi_{835}(12,\cdot)\)
\(\chi_{835}(18,\cdot)\)
\(\chi_{835}(22,\cdot)\)
\(\chi_{835}(27,\cdot)\)
\(\chi_{835}(28,\cdot)\)
\(\chi_{835}(32,\cdot)\)
\(\chi_{835}(33,\cdot)\)
\(\chi_{835}(38,\cdot)\)
\(\chi_{835}(42,\cdot)\)
\(\chi_{835}(47,\cdot)\)
\(\chi_{835}(48,\cdot)\)
\(\chi_{835}(57,\cdot)\)
\(\chi_{835}(58,\cdot)\)
\(\chi_{835}(62,\cdot)\)
\(\chi_{835}(63,\cdot)\)
\(\chi_{835}(72,\cdot)\)
\(\chi_{835}(77,\cdot)\)
\(\chi_{835}(87,\cdot)\)
\(\chi_{835}(88,\cdot)\)
\(\chi_{835}(93,\cdot)\)
\(\chi_{835}(97,\cdot)\)
\(\chi_{835}(98,\cdot)\)
\(\chi_{835}(107,\cdot)\)
\(\chi_{835}(108,\cdot)\)
\(\chi_{835}(112,\cdot)\)
\(\chi_{835}(122,\cdot)\)
...
![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 };
\((502,506)\) → \((-i,e\left(\frac{57}{83}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 835 }(128, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{73}{332}\right)\) | \(e\left(\frac{267}{332}\right)\) | \(e\left(\frac{73}{166}\right)\) | \(e\left(\frac{2}{83}\right)\) | \(e\left(\frac{261}{332}\right)\) | \(e\left(\frac{219}{332}\right)\) | \(e\left(\frac{101}{166}\right)\) | \(e\left(\frac{19}{83}\right)\) | \(e\left(\frac{81}{332}\right)\) | \(e\left(\frac{327}{332}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
gp:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
