![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2667, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([63,42,19]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1409, 2667))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2667.1409");
| Modulus: | \(2667\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2667\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(126\) |
![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);
|
\(\chi_{2667}(23,\cdot)\)
\(\chi_{2667}(53,\cdot)\)
\(\chi_{2667}(65,\cdot)\)
\(\chi_{2667}(116,\cdot)\)
\(\chi_{2667}(347,\cdot)\)
\(\chi_{2667}(410,\cdot)\)
\(\chi_{2667}(599,\cdot)\)
\(\chi_{2667}(863,\cdot)\)
\(\chi_{2667}(872,\cdot)\)
\(\chi_{2667}(998,\cdot)\)
\(\chi_{2667}(1061,\cdot)\)
\(\chi_{2667}(1073,\cdot)\)
\(\chi_{2667}(1157,\cdot)\)
\(\chi_{2667}(1199,\cdot)\)
\(\chi_{2667}(1229,\cdot)\)
\(\chi_{2667}(1313,\cdot)\)
\(\chi_{2667}(1355,\cdot)\)
\(\chi_{2667}(1367,\cdot)\)
\(\chi_{2667}(1376,\cdot)\)
\(\chi_{2667}(1409,\cdot)\)
\(\chi_{2667}(1493,\cdot)\)
\(\chi_{2667}(1607,\cdot)\)
\(\chi_{2667}(1817,\cdot)\)
\(\chi_{2667}(1892,\cdot)\)
\(\chi_{2667}(1997,\cdot)\)
\(\chi_{2667}(2039,\cdot)\)
\(\chi_{2667}(2090,\cdot)\)
\(\chi_{2667}(2144,\cdot)\)
\(\chi_{2667}(2165,\cdot)\)
\(\chi_{2667}(2207,\cdot)\)
...
![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: |
$\Q(\zeta_{63})$ |
![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 126 polynomial (not computed) |
![Copy content]()
sage:chi.fixed_field()
|
\((890,1144,2416)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{19}{126}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2667 }(1409, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{11}{126}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{71}{126}\right)\) | \(e\left(\frac{1}{3}\right)\) |
![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)
