![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(371, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([104,3]))
![Copy content]()
gp:[g,chi] = znchar(Mod(214, 371))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("371.214");
| Modulus: | \(371\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(371\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(156\) |
![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);
|
\(\chi_{371}(2,\cdot)\)
\(\chi_{371}(18,\cdot)\)
\(\chi_{371}(32,\cdot)\)
\(\chi_{371}(39,\cdot)\)
\(\chi_{371}(51,\cdot)\)
\(\chi_{371}(58,\cdot)\)
\(\chi_{371}(65,\cdot)\)
\(\chi_{371}(67,\cdot)\)
\(\chi_{371}(72,\cdot)\)
\(\chi_{371}(74,\cdot)\)
\(\chi_{371}(79,\cdot)\)
\(\chi_{371}(86,\cdot)\)
\(\chi_{371}(88,\cdot)\)
\(\chi_{371}(109,\cdot)\)
\(\chi_{371}(114,\cdot)\)
\(\chi_{371}(128,\cdot)\)
\(\chi_{371}(137,\cdot)\)
\(\chi_{371}(151,\cdot)\)
\(\chi_{371}(156,\cdot)\)
\(\chi_{371}(177,\cdot)\)
\(\chi_{371}(179,\cdot)\)
\(\chi_{371}(186,\cdot)\)
\(\chi_{371}(191,\cdot)\)
\(\chi_{371}(193,\cdot)\)
\(\chi_{371}(198,\cdot)\)
\(\chi_{371}(200,\cdot)\)
\(\chi_{371}(207,\cdot)\)
\(\chi_{371}(214,\cdot)\)
\(\chi_{371}(226,\cdot)\)
\(\chi_{371}(233,\cdot)\)
...
![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 };
\((213,267)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{52}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 371 }(214, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{55}{156}\right)\) | \(e\left(\frac{155}{156}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{37}{156}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{23}{39}\right)\) | \(e\left(\frac{61}{78}\right)\) | \(e\left(\frac{109}{156}\right)\) |
![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)
