![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(4800))
M = H._module
chi = DirichletCharacter(H, M([2400,2475,3696,1600]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1147, 416000))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.1147");
| Modulus: | \(416000\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(416000\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(4800\) |
![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);
|
\(\chi_{416000}(3,\cdot)\)
\(\chi_{416000}(347,\cdot)\)
\(\chi_{416000}(1147,\cdot)\)
\(\chi_{416000}(1283,\cdot)\)
\(\chi_{416000}(1387,\cdot)\)
![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: |
$\Q(\zeta_{4800})$ |
![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 4800 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((-1,e\left(\frac{33}{64}\right),e\left(\frac{77}{100}\right),e\left(\frac{1}{3}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(1147, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1297}{4800}\right)\) | \(e\left(\frac{371}{480}\right)\) | \(e\left(\frac{1297}{2400}\right)\) | \(e\left(\frac{871}{4800}\right)\) | \(e\left(\frac{377}{1200}\right)\) | \(e\left(\frac{4253}{4800}\right)\) | \(e\left(\frac{69}{1600}\right)\) | \(e\left(\frac{2213}{2400}\right)\) | \(e\left(\frac{1297}{1600}\right)\) | \(e\left(\frac{2377}{4800}\right)\) |
![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)
