![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(13671, base_ring=CyclotomicField(210))
M = H._module
chi = DirichletCharacter(H, M([105,185,42]))
![Copy content]()
gp:[g,chi] = znchar(Mod(7766, 13671))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("13671.7766");
| Modulus: | \(13671\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(4557\) |
![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: | no, induced from \(\chi_{4557}(3209,\cdot)\) |
![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);
|
\(\chi_{13671}(467,\cdot)\)
\(\chi_{13671}(593,\cdot)\)
\(\chi_{13671}(845,\cdot)\)
\(\chi_{13671}(1025,\cdot)\)
\(\chi_{13671}(1151,\cdot)\)
\(\chi_{13671}(1349,\cdot)\)
\(\chi_{13671}(1907,\cdot)\)
\(\chi_{13671}(2546,\cdot)\)
\(\chi_{13671}(2798,\cdot)\)
\(\chi_{13671}(2978,\cdot)\)
\(\chi_{13671}(3104,\cdot)\)
\(\chi_{13671}(3356,\cdot)\)
\(\chi_{13671}(3860,\cdot)\)
\(\chi_{13671}(4373,\cdot)\)
\(\chi_{13671}(4499,\cdot)\)
\(\chi_{13671}(4751,\cdot)\)
\(\chi_{13671}(5057,\cdot)\)
\(\chi_{13671}(5255,\cdot)\)
\(\chi_{13671}(5309,\cdot)\)
\(\chi_{13671}(6326,\cdot)\)
\(\chi_{13671}(6452,\cdot)\)
\(\chi_{13671}(6704,\cdot)\)
\(\chi_{13671}(6884,\cdot)\)
\(\chi_{13671}(7010,\cdot)\)
\(\chi_{13671}(7208,\cdot)\)
\(\chi_{13671}(7262,\cdot)\)
\(\chi_{13671}(7766,\cdot)\)
\(\chi_{13671}(8279,\cdot)\)
\(\chi_{13671}(8405,\cdot)\)
\(\chi_{13671}(8657,\cdot)\)
...
![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 };
\((6077,7255,7939)\) → \((-1,e\left(\frac{37}{42}\right),e\left(\frac{1}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 13671 }(7766, a) \) |
\(1\) | \(1\) | \(e\left(\frac{43}{210}\right)\) | \(e\left(\frac{43}{105}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{53}{210}\right)\) | \(e\left(\frac{71}{210}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{86}{105}\right)\) | \(e\left(\frac{97}{105}\right)\) | \(e\left(\frac{19}{30}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
