![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2989, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([95,196]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1018, 2989))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2989.1018");
| Modulus: | \(2989\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2989\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{2989}(73,\cdot)\)
\(\chi_{2989}(138,\cdot)\)
\(\chi_{2989}(164,\cdot)\)
\(\chi_{2989}(269,\cdot)\)
\(\chi_{2989}(320,\cdot)\)
\(\chi_{2989}(327,\cdot)\)
\(\chi_{2989}(500,\cdot)\)
\(\chi_{2989}(544,\cdot)\)
\(\chi_{2989}(565,\cdot)\)
\(\chi_{2989}(591,\cdot)\)
\(\chi_{2989}(696,\cdot)\)
\(\chi_{2989}(747,\cdot)\)
\(\chi_{2989}(850,\cdot)\)
\(\chi_{2989}(927,\cdot)\)
\(\chi_{2989}(971,\cdot)\)
\(\chi_{2989}(992,\cdot)\)
\(\chi_{2989}(1018,\cdot)\)
\(\chi_{2989}(1123,\cdot)\)
\(\chi_{2989}(1174,\cdot)\)
\(\chi_{2989}(1181,\cdot)\)
\(\chi_{2989}(1277,\cdot)\)
\(\chi_{2989}(1398,\cdot)\)
\(\chi_{2989}(1419,\cdot)\)
\(\chi_{2989}(1445,\cdot)\)
\(\chi_{2989}(1601,\cdot)\)
\(\chi_{2989}(1608,\cdot)\)
\(\chi_{2989}(1704,\cdot)\)
\(\chi_{2989}(1781,\cdot)\)
\(\chi_{2989}(1825,\cdot)\)
\(\chi_{2989}(1846,\cdot)\)
...
![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 };
\((2502,246)\) → \((e\left(\frac{19}{42}\right),e\left(\frac{14}{15}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 2989 }(1018, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{73}{105}\right)\) | \(e\left(\frac{11}{210}\right)\) | \(e\left(\frac{41}{105}\right)\) | \(e\left(\frac{137}{210}\right)\) | \(e\left(\frac{157}{210}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{11}{105}\right)\) | \(e\left(\frac{73}{210}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{31}{70}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
