![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4189, base_ring=CyclotomicField(2030))
M = H._module
chi = DirichletCharacter(H, M([1470,1247]))
![Copy content]()
gp:[g,chi] = znchar(Mod(186, 4189))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4189.186");
| Modulus: | \(4189\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(4189\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(2030\) |
![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);
|
\(\chi_{4189}(7,\cdot)\)
\(\chi_{4189}(21,\cdot)\)
\(\chi_{4189}(22,\cdot)\)
\(\chi_{4189}(28,\cdot)\)
\(\chi_{4189}(35,\cdot)\)
\(\chi_{4189}(53,\cdot)\)
\(\chi_{4189}(62,\cdot)\)
\(\chi_{4189}(63,\cdot)\)
\(\chi_{4189}(68,\cdot)\)
\(\chi_{4189}(78,\cdot)\)
\(\chi_{4189}(84,\cdot)\)
\(\chi_{4189}(104,\cdot)\)
\(\chi_{4189}(123,\cdot)\)
\(\chi_{4189}(127,\cdot)\)
\(\chi_{4189}(130,\cdot)\)
\(\chi_{4189}(133,\cdot)\)
\(\chi_{4189}(134,\cdot)\)
\(\chi_{4189}(138,\cdot)\)
\(\chi_{4189}(139,\cdot)\)
\(\chi_{4189}(140,\cdot)\)
\(\chi_{4189}(153,\cdot)\)
\(\chi_{4189}(163,\cdot)\)
\(\chi_{4189}(164,\cdot)\)
\(\chi_{4189}(175,\cdot)\)
\(\chi_{4189}(184,\cdot)\)
\(\chi_{4189}(186,\cdot)\)
\(\chi_{4189}(189,\cdot)\)
\(\chi_{4189}(194,\cdot)\)
\(\chi_{4189}(197,\cdot)\)
\(\chi_{4189}(198,\cdot)\)
...
![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 };
\((356,2066)\) → \((e\left(\frac{21}{29}\right),e\left(\frac{43}{70}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 4189 }(186, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{416}{1015}\right)\) | \(e\left(\frac{181}{1015}\right)\) | \(e\left(\frac{832}{1015}\right)\) | \(e\left(\frac{79}{145}\right)\) | \(e\left(\frac{597}{1015}\right)\) | \(e\left(\frac{1317}{2030}\right)\) | \(e\left(\frac{233}{1015}\right)\) | \(e\left(\frac{362}{1015}\right)\) | \(e\left(\frac{969}{1015}\right)\) | \(e\left(\frac{297}{2030}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
