![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9025, base_ring=CyclotomicField(1710))
M = H._module
chi = DirichletCharacter(H, M([1539,520]))
![Copy content]()
gp:[g,chi] = znchar(Mod(44, 9025))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("9025.44");
| Modulus: | \(9025\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(9025\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(1710\) |
![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);
|
\(\chi_{9025}(4,\cdot)\)
\(\chi_{9025}(9,\cdot)\)
\(\chi_{9025}(44,\cdot)\)
\(\chi_{9025}(104,\cdot)\)
\(\chi_{9025}(119,\cdot)\)
\(\chi_{9025}(139,\cdot)\)
\(\chi_{9025}(169,\cdot)\)
\(\chi_{9025}(194,\cdot)\)
\(\chi_{9025}(214,\cdot)\)
\(\chi_{9025}(244,\cdot)\)
\(\chi_{9025}(264,\cdot)\)
\(\chi_{9025}(289,\cdot)\)
\(\chi_{9025}(294,\cdot)\)
\(\chi_{9025}(309,\cdot)\)
\(\chi_{9025}(329,\cdot)\)
\(\chi_{9025}(339,\cdot)\)
\(\chi_{9025}(359,\cdot)\)
\(\chi_{9025}(384,\cdot)\)
\(\chi_{9025}(404,\cdot)\)
\(\chi_{9025}(434,\cdot)\)
\(\chi_{9025}(454,\cdot)\)
\(\chi_{9025}(479,\cdot)\)
\(\chi_{9025}(484,\cdot)\)
\(\chi_{9025}(519,\cdot)\)
\(\chi_{9025}(529,\cdot)\)
\(\chi_{9025}(579,\cdot)\)
\(\chi_{9025}(594,\cdot)\)
\(\chi_{9025}(614,\cdot)\)
\(\chi_{9025}(644,\cdot)\)
\(\chi_{9025}(669,\cdot)\)
...
![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: |
$\Q(\zeta_{855})$ |
![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 1710 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((5777,3251)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{52}{171}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 9025 }(44, a) \) |
\(1\) | \(1\) | \(e\left(\frac{349}{1710}\right)\) | \(e\left(\frac{973}{1710}\right)\) | \(e\left(\frac{349}{855}\right)\) | \(e\left(\frac{661}{855}\right)\) | \(e\left(\frac{13}{114}\right)\) | \(e\left(\frac{349}{570}\right)\) | \(e\left(\frac{118}{855}\right)\) | \(e\left(\frac{119}{285}\right)\) | \(e\left(\frac{557}{570}\right)\) | \(e\left(\frac{251}{1710}\right)\) |
![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)
