![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13351, base_ring=CyclotomicField(156))
M = H._module
chi = DirichletCharacter(H, M([19,74]))
![Copy content]()
gp:[g,chi] = znchar(Mod(2247, 13351))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13351.2247");
| Modulus: | \(13351\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(13351\) |
![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);
|
\(\chi_{13351}(37,\cdot)\)
\(\chi_{13351}(63,\cdot)\)
\(\chi_{13351}(592,\cdot)\)
\(\chi_{13351}(1008,\cdot)\)
\(\chi_{13351}(2247,\cdot)\)
\(\chi_{13351}(2368,\cdot)\)
\(\chi_{13351}(2399,\cdot)\)
\(\chi_{13351}(2438,\cdot)\)
\(\chi_{13351}(2650,\cdot)\)
\(\chi_{13351}(3309,\cdot)\)
\(\chi_{13351}(3347,\cdot)\)
\(\chi_{13351}(4032,\cdot)\)
\(\chi_{13351}(4379,\cdot)\)
\(\chi_{13351}(4556,\cdot)\)
\(\chi_{13351}(4691,\cdot)\)
\(\chi_{13351}(4799,\cdot)\)
\(\chi_{13351}(4873,\cdot)\)
\(\chi_{13351}(5510,\cdot)\)
\(\chi_{13351}(5514,\cdot)\)
\(\chi_{13351}(5770,\cdot)\)
\(\chi_{13351}(5991,\cdot)\)
\(\chi_{13351}(6675,\cdot)\)
\(\chi_{13351}(7338,\cdot)\)
\(\chi_{13351}(8053,\cdot)\)
\(\chi_{13351}(8054,\cdot)\)
\(\chi_{13351}(8118,\cdot)\)
\(\chi_{13351}(8301,\cdot)\)
\(\chi_{13351}(8803,\cdot)\)
\(\chi_{13351}(9171,\cdot)\)
\(\chi_{13351}(9250,\cdot)\)
...
![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 };
\((7269,12169)\) → \((e\left(\frac{19}{156}\right),e\left(\frac{37}{78}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 13351 }(2247, a) \) |
\(1\) | \(1\) | \(e\left(\frac{1}{52}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{79}{156}\right)\) | \(e\left(\frac{31}{52}\right)\) | \(e\left(\frac{9}{52}\right)\) | \(e\left(\frac{3}{52}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{41}{78}\right)\) | \(e\left(\frac{125}{156}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
