![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1625, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([21,25]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1003, 1625))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1625.1003");
| Modulus: | \(1625\) |
![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: | 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);
|
\(\chi_{1625}(28,\cdot)\)
\(\chi_{1625}(37,\cdot)\)
\(\chi_{1625}(58,\cdot)\)
\(\chi_{1625}(72,\cdot)\)
\(\chi_{1625}(102,\cdot)\)
\(\chi_{1625}(123,\cdot)\)
\(\chi_{1625}(137,\cdot)\)
\(\chi_{1625}(158,\cdot)\)
\(\chi_{1625}(167,\cdot)\)
\(\chi_{1625}(188,\cdot)\)
\(\chi_{1625}(202,\cdot)\)
\(\chi_{1625}(223,\cdot)\)
\(\chi_{1625}(253,\cdot)\)
\(\chi_{1625}(267,\cdot)\)
\(\chi_{1625}(288,\cdot)\)
\(\chi_{1625}(297,\cdot)\)
\(\chi_{1625}(353,\cdot)\)
\(\chi_{1625}(362,\cdot)\)
\(\chi_{1625}(383,\cdot)\)
\(\chi_{1625}(397,\cdot)\)
\(\chi_{1625}(427,\cdot)\)
\(\chi_{1625}(448,\cdot)\)
\(\chi_{1625}(462,\cdot)\)
\(\chi_{1625}(483,\cdot)\)
\(\chi_{1625}(492,\cdot)\)
\(\chi_{1625}(513,\cdot)\)
\(\chi_{1625}(527,\cdot)\)
\(\chi_{1625}(548,\cdot)\)
\(\chi_{1625}(578,\cdot)\)
\(\chi_{1625}(592,\cdot)\)
...
![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()
|
\((1002,626)\) → \((e\left(\frac{7}{100}\right),e\left(\frac{1}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 1625 }(1003, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{150}\right)\) | \(e\left(\frac{247}{300}\right)\) | \(e\left(\frac{23}{75}\right)\) | \(e\left(\frac{293}{300}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{50}\right)\) | \(e\left(\frac{97}{150}\right)\) | \(e\left(\frac{271}{300}\right)\) | \(e\left(\frac{13}{100}\right)\) | \(e\left(\frac{1}{50}\right)\) |
![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)
