![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(83200, base_ring=CyclotomicField(320))
M = H._module
chi = DirichletCharacter(H, M([160,145,256,160]))
![Copy content]()
gp:[g,chi] = znchar(Mod(74411, 83200))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("83200.74411");
| Modulus: | \(83200\) |
![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: | 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);
|
\(\chi_{83200}(571,\cdot)\)
\(\chi_{83200}(1091,\cdot)\)
\(\chi_{83200}(1611,\cdot)\)
\(\chi_{83200}(2131,\cdot)\)
\(\chi_{83200}(3171,\cdot)\)
\(\chi_{83200}(3691,\cdot)\)
\(\chi_{83200}(4211,\cdot)\)
\(\chi_{83200}(4731,\cdot)\)
\(\chi_{83200}(5771,\cdot)\)
\(\chi_{83200}(6291,\cdot)\)
\(\chi_{83200}(6811,\cdot)\)
\(\chi_{83200}(7331,\cdot)\)
\(\chi_{83200}(8371,\cdot)\)
\(\chi_{83200}(8891,\cdot)\)
\(\chi_{83200}(9411,\cdot)\)
\(\chi_{83200}(9931,\cdot)\)
\(\chi_{83200}(10971,\cdot)\)
\(\chi_{83200}(11491,\cdot)\)
\(\chi_{83200}(12011,\cdot)\)
\(\chi_{83200}(12531,\cdot)\)
\(\chi_{83200}(13571,\cdot)\)
\(\chi_{83200}(14091,\cdot)\)
\(\chi_{83200}(14611,\cdot)\)
\(\chi_{83200}(15131,\cdot)\)
\(\chi_{83200}(16171,\cdot)\)
\(\chi_{83200}(16691,\cdot)\)
\(\chi_{83200}(17211,\cdot)\)
\(\chi_{83200}(17731,\cdot)\)
\(\chi_{83200}(18771,\cdot)\)
\(\chi_{83200}(19291,\cdot)\)
...
![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()
|
\((74751,16901,56577,64001)\) → \((-1,e\left(\frac{29}{64}\right),e\left(\frac{4}{5}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 83200 }(74411, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{307}{320}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{147}{160}\right)\) | \(e\left(\frac{101}{320}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{263}{320}\right)\) | \(e\left(\frac{157}{320}\right)\) | \(e\left(\frac{103}{160}\right)\) | \(e\left(\frac{281}{320}\right)\) | \(e\left(\frac{107}{320}\right)\) |
![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)
