![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6500, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([0,219,50]))
![Copy content]()
gp:[g,chi] = znchar(Mod(17, 6500))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6500.17");
| Modulus: | \(6500\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(1625\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(300\) |
![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 \(\chi_{1625}(17,\cdot)\) |
![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);
|
\(\chi_{6500}(17,\cdot)\)
\(\chi_{6500}(153,\cdot)\)
\(\chi_{6500}(173,\cdot)\)
\(\chi_{6500}(277,\cdot)\)
\(\chi_{6500}(413,\cdot)\)
\(\chi_{6500}(433,\cdot)\)
\(\chi_{6500}(517,\cdot)\)
\(\chi_{6500}(537,\cdot)\)
\(\chi_{6500}(673,\cdot)\)
\(\chi_{6500}(777,\cdot)\)
\(\chi_{6500}(797,\cdot)\)
\(\chi_{6500}(933,\cdot)\)
\(\chi_{6500}(953,\cdot)\)
\(\chi_{6500}(1037,\cdot)\)
\(\chi_{6500}(1213,\cdot)\)
\(\chi_{6500}(1297,\cdot)\)
\(\chi_{6500}(1317,\cdot)\)
\(\chi_{6500}(1453,\cdot)\)
\(\chi_{6500}(1473,\cdot)\)
\(\chi_{6500}(1577,\cdot)\)
\(\chi_{6500}(1713,\cdot)\)
\(\chi_{6500}(1733,\cdot)\)
\(\chi_{6500}(1817,\cdot)\)
\(\chi_{6500}(1837,\cdot)\)
\(\chi_{6500}(1973,\cdot)\)
\(\chi_{6500}(2077,\cdot)\)
\(\chi_{6500}(2097,\cdot)\)
\(\chi_{6500}(2233,\cdot)\)
\(\chi_{6500}(2253,\cdot)\)
\(\chi_{6500}(2337,\cdot)\)
...
![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: |
$\Q(\zeta_{300})$ |
![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 300 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((3251,5877,5501)\) → \((1,e\left(\frac{73}{100}\right),e\left(\frac{1}{6}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6500 }(17, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{233}{300}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{83}{150}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{187}{300}\right)\) | \(e\left(\frac{73}{75}\right)\) | \(e\left(\frac{33}{50}\right)\) | \(e\left(\frac{89}{300}\right)\) | \(e\left(\frac{33}{100}\right)\) | \(e\left(\frac{139}{150}\right)\) |
![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)
