![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6400, base_ring=CyclotomicField(320))
M = H._module
chi = DirichletCharacter(H, M([160,165,272]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1147, 6400))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6400.1147");
| Modulus: | \(6400\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(6400\) |
![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: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{6400}(3,\cdot)\)
\(\chi_{6400}(27,\cdot)\)
\(\chi_{6400}(83,\cdot)\)
\(\chi_{6400}(163,\cdot)\)
\(\chi_{6400}(187,\cdot)\)
\(\chi_{6400}(267,\cdot)\)
\(\chi_{6400}(323,\cdot)\)
\(\chi_{6400}(347,\cdot)\)
\(\chi_{6400}(403,\cdot)\)
\(\chi_{6400}(427,\cdot)\)
\(\chi_{6400}(483,\cdot)\)
\(\chi_{6400}(563,\cdot)\)
\(\chi_{6400}(587,\cdot)\)
\(\chi_{6400}(667,\cdot)\)
\(\chi_{6400}(723,\cdot)\)
\(\chi_{6400}(747,\cdot)\)
\(\chi_{6400}(803,\cdot)\)
\(\chi_{6400}(827,\cdot)\)
\(\chi_{6400}(883,\cdot)\)
\(\chi_{6400}(963,\cdot)\)
\(\chi_{6400}(987,\cdot)\)
\(\chi_{6400}(1067,\cdot)\)
\(\chi_{6400}(1123,\cdot)\)
\(\chi_{6400}(1147,\cdot)\)
\(\chi_{6400}(1203,\cdot)\)
\(\chi_{6400}(1227,\cdot)\)
\(\chi_{6400}(1283,\cdot)\)
\(\chi_{6400}(1363,\cdot)\)
\(\chi_{6400}(1387,\cdot)\)
\(\chi_{6400}(1467,\cdot)\)
...
![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()
|
\((4351,4101,5377)\) → \((-1,e\left(\frac{33}{64}\right),e\left(\frac{17}{20}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 6400 }(1147, a) \) |
\(1\) | \(1\) | \(e\left(\frac{159}{320}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{159}{160}\right)\) | \(e\left(\frac{297}{320}\right)\) | \(e\left(\frac{123}{320}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{211}{320}\right)\) | \(e\left(\frac{129}{320}\right)\) | \(e\left(\frac{11}{160}\right)\) | \(e\left(\frac{157}{320}\right)\) |
![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)
