![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16709, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([75,168,119]))
![Copy content]()
gp:[g,chi] = znchar(Mod(9694, 16709))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16709.9694");
| Modulus: | \(16709\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(16709\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(210\) |
![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);
|
\(\chi_{16709}(916,\cdot)\)
\(\chi_{16709}(972,\cdot)\)
\(\chi_{16709}(1252,\cdot)\)
\(\chi_{16709}(1336,\cdot)\)
\(\chi_{16709}(1357,\cdot)\)
\(\chi_{16709}(2183,\cdot)\)
\(\chi_{16709}(2533,\cdot)\)
\(\chi_{16709}(3303,\cdot)\)
\(\chi_{16709}(3359,\cdot)\)
\(\chi_{16709}(3639,\cdot)\)
\(\chi_{16709}(3744,\cdot)\)
\(\chi_{16709}(4570,\cdot)\)
\(\chi_{16709}(4640,\cdot)\)
\(\chi_{16709}(4920,\cdot)\)
\(\chi_{16709}(5690,\cdot)\)
\(\chi_{16709}(5746,\cdot)\)
\(\chi_{16709}(6110,\cdot)\)
\(\chi_{16709}(6131,\cdot)\)
\(\chi_{16709}(7027,\cdot)\)
\(\chi_{16709}(7307,\cdot)\)
\(\chi_{16709}(8077,\cdot)\)
\(\chi_{16709}(8413,\cdot)\)
\(\chi_{16709}(8497,\cdot)\)
\(\chi_{16709}(8518,\cdot)\)
\(\chi_{16709}(9344,\cdot)\)
\(\chi_{16709}(9414,\cdot)\)
\(\chi_{16709}(9694,\cdot)\)
\(\chi_{16709}(10464,\cdot)\)
\(\chi_{16709}(10520,\cdot)\)
\(\chi_{16709}(10800,\cdot)\)
...
![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 };
\((16369,1520,14015)\) → \((e\left(\frac{5}{14}\right),e\left(\frac{4}{5}\right),e\left(\frac{17}{30}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \( \chi_{ 16709 }(9694, a) \) |
\(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{34}{105}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{187}{210}\right)\) | \(e\left(\frac{1}{105}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{68}{105}\right)\) | \(e\left(\frac{121}{210}\right)\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{86}{105}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
