![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(41600, base_ring=CyclotomicField(480))
M = H._module
chi = DirichletCharacter(H, M([0,285,384,40]))
![Copy content]()
gp:[g,chi] = znchar(Mod(9661, 41600))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("41600.9661");
| Modulus: | \(41600\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(41600\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(480\) |
![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);
|
\(\chi_{41600}(461,\cdot)\)
\(\chi_{41600}(661,\cdot)\)
\(\chi_{41600}(1341,\cdot)\)
\(\chi_{41600}(1541,\cdot)\)
\(\chi_{41600}(2381,\cdot)\)
\(\chi_{41600}(2541,\cdot)\)
\(\chi_{41600}(2581,\cdot)\)
\(\chi_{41600}(2741,\cdot)\)
\(\chi_{41600}(3421,\cdot)\)
\(\chi_{41600}(3581,\cdot)\)
\(\chi_{41600}(3621,\cdot)\)
\(\chi_{41600}(3781,\cdot)\)
\(\chi_{41600}(4461,\cdot)\)
\(\chi_{41600}(4621,\cdot)\)
\(\chi_{41600}(4661,\cdot)\)
\(\chi_{41600}(4821,\cdot)\)
\(\chi_{41600}(5661,\cdot)\)
\(\chi_{41600}(5861,\cdot)\)
\(\chi_{41600}(6541,\cdot)\)
\(\chi_{41600}(6741,\cdot)\)
\(\chi_{41600}(7581,\cdot)\)
\(\chi_{41600}(7741,\cdot)\)
\(\chi_{41600}(7781,\cdot)\)
\(\chi_{41600}(7941,\cdot)\)
\(\chi_{41600}(8621,\cdot)\)
\(\chi_{41600}(8781,\cdot)\)
\(\chi_{41600}(8821,\cdot)\)
\(\chi_{41600}(8981,\cdot)\)
\(\chi_{41600}(9661,\cdot)\)
\(\chi_{41600}(9821,\cdot)\)
...
![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: |
$\Q(\zeta_{480})$ |
![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 480 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((33151,16901,14977,22401)\) → \((1,e\left(\frac{19}{32}\right),e\left(\frac{4}{5}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 41600 }(9661, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{343}{480}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{103}{240}\right)\) | \(e\left(\frac{409}{480}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{227}{480}\right)\) | \(e\left(\frac{91}{160}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{23}{160}\right)\) | \(e\left(\frac{463}{480}\right)\) |
![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)
