![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(416000, base_ring=CyclotomicField(320))
M = H._module
chi = DirichletCharacter(H, M([160,315,256,160]))
![Copy content]()
gp:[g,chi] = znchar(Mod(51, 416000))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("416000.51");
| Modulus: | \(416000\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(83200\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(320\) |
![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: | no, induced from \(\chi_{83200}(66611,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | no |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{416000}(51,\cdot)\)
\(\chi_{416000}(2651,\cdot)\)
\(\chi_{416000}(7851,\cdot)\)
\(\chi_{416000}(10451,\cdot)\)
\(\chi_{416000}(13051,\cdot)\)
\(\chi_{416000}(15651,\cdot)\)
\(\chi_{416000}(20851,\cdot)\)
\(\chi_{416000}(23451,\cdot)\)
\(\chi_{416000}(26051,\cdot)\)
\(\chi_{416000}(28651,\cdot)\)
\(\chi_{416000}(33851,\cdot)\)
\(\chi_{416000}(36451,\cdot)\)
\(\chi_{416000}(39051,\cdot)\)
\(\chi_{416000}(41651,\cdot)\)
\(\chi_{416000}(46851,\cdot)\)
\(\chi_{416000}(49451,\cdot)\)
\(\chi_{416000}(52051,\cdot)\)
\(\chi_{416000}(54651,\cdot)\)
\(\chi_{416000}(59851,\cdot)\)
\(\chi_{416000}(62451,\cdot)\)
\(\chi_{416000}(65051,\cdot)\)
\(\chi_{416000}(67651,\cdot)\)
\(\chi_{416000}(72851,\cdot)\)
\(\chi_{416000}(75451,\cdot)\)
\(\chi_{416000}(78051,\cdot)\)
\(\chi_{416000}(80651,\cdot)\)
\(\chi_{416000}(85851,\cdot)\)
\(\chi_{416000}(88451,\cdot)\)
\(\chi_{416000}(91051,\cdot)\)
\(\chi_{416000}(93651,\cdot)\)
...
![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: |
$\Q(\zeta_{320})$ |
![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 320 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((74751,266501,389377,64001)\) → \((-1,e\left(\frac{63}{64}\right),e\left(\frac{4}{5}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 416000 }(51, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{177}{320}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{17}{160}\right)\) | \(e\left(\frac{151}{320}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{13}{320}\right)\) | \(e\left(\frac{127}{320}\right)\) | \(e\left(\frac{13}{160}\right)\) | \(e\left(\frac{211}{320}\right)\) | \(e\left(\frac{217}{320}\right)\) |
![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)
