![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([80,87]))
![Copy content]()
gp:[g,chi] = znchar(Mod(186, 287))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("287.186");
| Modulus: | \(287\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(287\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(120\) |
![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);
|
\(\chi_{287}(11,\cdot)\)
\(\chi_{287}(30,\cdot)\)
\(\chi_{287}(53,\cdot)\)
\(\chi_{287}(58,\cdot)\)
\(\chi_{287}(60,\cdot)\)
\(\chi_{287}(65,\cdot)\)
\(\chi_{287}(67,\cdot)\)
\(\chi_{287}(88,\cdot)\)
\(\chi_{287}(93,\cdot)\)
\(\chi_{287}(95,\cdot)\)
\(\chi_{287}(116,\cdot)\)
\(\chi_{287}(130,\cdot)\)
\(\chi_{287}(135,\cdot)\)
\(\chi_{287}(142,\cdot)\)
\(\chi_{287}(149,\cdot)\)
\(\chi_{287}(151,\cdot)\)
\(\chi_{287}(158,\cdot)\)
\(\chi_{287}(170,\cdot)\)
\(\chi_{287}(177,\cdot)\)
\(\chi_{287}(179,\cdot)\)
\(\chi_{287}(186,\cdot)\)
\(\chi_{287}(193,\cdot)\)
\(\chi_{287}(198,\cdot)\)
\(\chi_{287}(212,\cdot)\)
\(\chi_{287}(233,\cdot)\)
\(\chi_{287}(235,\cdot)\)
\(\chi_{287}(240,\cdot)\)
\(\chi_{287}(261,\cdot)\)
\(\chi_{287}(263,\cdot)\)
\(\chi_{287}(268,\cdot)\)
...
![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 };
| Field of values: |
$\Q(\zeta_{120})$ |
![Copy content]()
sage:CyclotomicField(chi.multiplicative_order())
![Copy content]()
gp:nfinit(polcyclo(charorder(g,chi)))
![Copy content]()
magma:CyclotomicField(Order(chi));
|
| Fixed field: |
Number field defined by a degree 120 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((206,211)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{29}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 287 }(186, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{109}{120}\right)\) |
![Copy content]()
sage:chi(x) # x integer
![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)
