![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15867, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([80,117,100]))
![Copy content]()
gp:[g,chi] = znchar(Mod(7, 15867))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15867.7");
| Modulus: | \(15867\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(15867\) |
![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: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{15867}(7,\cdot)\)
\(\chi_{15867}(1327,\cdot)\)
\(\chi_{15867}(1942,\cdot)\)
\(\chi_{15867}(2488,\cdot)\)
\(\chi_{15867}(3103,\cdot)\)
\(\chi_{15867}(4810,\cdot)\)
\(\chi_{15867}(5425,\cdot)\)
\(\chi_{15867}(5971,\cdot)\)
\(\chi_{15867}(6586,\cdot)\)
\(\chi_{15867}(7906,\cdot)\)
\(\chi_{15867}(8293,\cdot)\)
\(\chi_{15867}(8521,\cdot)\)
\(\chi_{15867}(8680,\cdot)\)
\(\chi_{15867}(8908,\cdot)\)
\(\chi_{15867}(9067,\cdot)\)
\(\chi_{15867}(9295,\cdot)\)
\(\chi_{15867}(9454,\cdot)\)
\(\chi_{15867}(9682,\cdot)\)
\(\chi_{15867}(10069,\cdot)\)
\(\chi_{15867}(10228,\cdot)\)
\(\chi_{15867}(10843,\cdot)\)
\(\chi_{15867}(12937,\cdot)\)
\(\chi_{15867}(13552,\cdot)\)
\(\chi_{15867}(13711,\cdot)\)
\(\chi_{15867}(14098,\cdot)\)
\(\chi_{15867}(14326,\cdot)\)
\(\chi_{15867}(14485,\cdot)\)
\(\chi_{15867}(14713,\cdot)\)
\(\chi_{15867}(14872,\cdot)\)
\(\chi_{15867}(15100,\cdot)\)
...
![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()
|
\((14105,15094,7012)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{39}{40}\right),e\left(\frac{5}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 15867 }(7, a) \) |
\(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{15}\right)\) |
![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)
