![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(15785, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([90,20,48,33]))
![Copy content]()
gp:[g,chi] = znchar(Mod(9663, 15785))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("15785.9663");
| Modulus: | \(15785\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(15785\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(120\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{15785}(423,\cdot)\)
\(\chi_{15785}(1347,\cdot)\)
\(\chi_{15785}(3743,\cdot)\)
\(\chi_{15785}(4933,\cdot)\)
\(\chi_{15785}(5218,\cdot)\)
\(\chi_{15785}(5267,\cdot)\)
\(\chi_{15785}(5582,\cdot)\)
\(\chi_{15785}(5757,\cdot)\)
\(\chi_{15785}(5857,\cdot)\)
\(\chi_{15785}(6548,\cdot)\)
\(\chi_{15785}(6912,\cdot)\)
\(\chi_{15785}(7857,\cdot)\)
\(\chi_{15785}(8193,\cdot)\)
\(\chi_{15785}(9663,\cdot)\)
\(\chi_{15785}(9728,\cdot)\)
\(\chi_{15785}(9777,\cdot)\)
\(\chi_{15785}(10092,\cdot)\)
\(\chi_{15785}(10267,\cdot)\)
\(\chi_{15785}(10272,\cdot)\)
\(\chi_{15785}(10503,\cdot)\)
\(\chi_{15785}(10552,\cdot)\)
\(\chi_{15785}(10608,\cdot)\)
\(\chi_{15785}(11058,\cdot)\)
\(\chi_{15785}(11422,\cdot)\)
\(\chi_{15785}(12367,\cdot)\)
\(\chi_{15785}(12703,\cdot)\)
\(\chi_{15785}(14173,\cdot)\)
\(\chi_{15785}(14782,\cdot)\)
\(\chi_{15785}(15013,\cdot)\)
\(\chi_{15785}(15018,\cdot)\)
...
![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 };
\((9472,4511,12916,3081)\) → \((-i,e\left(\frac{1}{6}\right),e\left(\frac{2}{5}\right),e\left(\frac{11}{40}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
| \( \chi_{ 15785 }(9663, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{71}{120}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
