![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66755, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([39,146,138]))
![Copy content]()
gp:[g,chi] = znchar(Mod(43957, 66755))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("66755.43957");
| Modulus: | \(66755\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(66755\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(156\) |
![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);
|
\(\chi_{66755}(927,\cdot)\)
\(\chi_{66755}(2032,\cdot)\)
\(\chi_{66755}(3098,\cdot)\)
\(\chi_{66755}(3488,\cdot)\)
\(\chi_{66755}(9968,\cdot)\)
\(\chi_{66755}(10917,\cdot)\)
\(\chi_{66755}(12178,\cdot)\)
\(\chi_{66755}(14278,\cdot)\)
\(\chi_{66755}(15383,\cdot)\)
\(\chi_{66755}(18977,\cdot)\)
\(\chi_{66755}(19887,\cdot)\)
\(\chi_{66755}(21837,\cdot)\)
\(\chi_{66755}(23137,\cdot)\)
\(\chi_{66755}(24268,\cdot)\)
\(\chi_{66755}(27642,\cdot)\)
\(\chi_{66755}(28467,\cdot)\)
\(\chi_{66755}(30047,\cdot)\)
\(\chi_{66755}(32328,\cdot)\)
\(\chi_{66755}(33238,\cdot)\)
\(\chi_{66755}(35188,\cdot)\)
\(\chi_{66755}(36488,\cdot)\)
\(\chi_{66755}(38672,\cdot)\)
\(\chi_{66755}(39277,\cdot)\)
\(\chi_{66755}(40993,\cdot)\)
\(\chi_{66755}(41467,\cdot)\)
\(\chi_{66755}(41818,\cdot)\)
\(\chi_{66755}(43307,\cdot)\)
\(\chi_{66755}(43398,\cdot)\)
\(\chi_{66755}(43957,\cdot)\)
\(\chi_{66755}(48962,\cdot)\)
...
![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 };
\((13352,33971,38871)\) → \((i,e\left(\frac{73}{78}\right),e\left(\frac{23}{26}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 66755 }(43957, a) \) |
\(1\) | \(1\) | \(e\left(\frac{113}{156}\right)\) | \(e\left(\frac{107}{156}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{16}{39}\right)\) | \(e\left(\frac{43}{156}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{7}{52}\right)\) | \(1\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
