![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4251, base_ring=CyclotomicField(108))
M = H._module
chi = DirichletCharacter(H, M([54,63,92]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1220, 4251))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4251.1220");
| Modulus: | \(4251\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(4251\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(108\) |
![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);
|
\(\chi_{4251}(158,\cdot)\)
\(\chi_{4251}(266,\cdot)\)
\(\chi_{4251}(332,\cdot)\)
\(\chi_{4251}(362,\cdot)\)
\(\chi_{4251}(509,\cdot)\)
\(\chi_{4251}(734,\cdot)\)
\(\chi_{4251}(1007,\cdot)\)
\(\chi_{4251}(1016,\cdot)\)
\(\chi_{4251}(1202,\cdot)\)
\(\chi_{4251}(1220,\cdot)\)
\(\chi_{4251}(1280,\cdot)\)
\(\chi_{4251}(1397,\cdot)\)
\(\chi_{4251}(1424,\cdot)\)
\(\chi_{4251}(1514,\cdot)\)
\(\chi_{4251}(1766,\cdot)\)
\(\chi_{4251}(1874,\cdot)\)
\(\chi_{4251}(2195,\cdot)\)
\(\chi_{4251}(2294,\cdot)\)
\(\chi_{4251}(2420,\cdot)\)
\(\chi_{4251}(2555,\cdot)\)
\(\chi_{4251}(2585,\cdot)\)
\(\chi_{4251}(2849,\cdot)\)
\(\chi_{4251}(3023,\cdot)\)
\(\chi_{4251}(3170,\cdot)\)
\(\chi_{4251}(3239,\cdot)\)
\(\chi_{4251}(3296,\cdot)\)
\(\chi_{4251}(3386,\cdot)\)
\(\chi_{4251}(3404,\cdot)\)
\(\chi_{4251}(3491,\cdot)\)
\(\chi_{4251}(3569,\cdot)\)
...
![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 };
\((1418,2290,2731)\) → \((-1,e\left(\frac{7}{12}\right),e\left(\frac{23}{27}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 4251 }(1220, a) \) |
\(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{53}{108}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{31}{108}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
