![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33813, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([51,68,6]))
![Copy content]()
gp:[g,chi] = znchar(Mod(25892, 33813))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33813.25892");
| Modulus: | \(33813\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(11271\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(102\) |
![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 \(\chi_{11271}(3350,\cdot)\) |
![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);
|
\(\chi_{33813}(35,\cdot)\)
\(\chi_{33813}(341,\cdot)\)
\(\chi_{33813}(2330,\cdot)\)
\(\chi_{33813}(4013,\cdot)\)
\(\chi_{33813}(4319,\cdot)\)
\(\chi_{33813}(6002,\cdot)\)
\(\chi_{33813}(6308,\cdot)\)
\(\chi_{33813}(7991,\cdot)\)
\(\chi_{33813}(8297,\cdot)\)
\(\chi_{33813}(9980,\cdot)\)
\(\chi_{33813}(10286,\cdot)\)
\(\chi_{33813}(11969,\cdot)\)
\(\chi_{33813}(12275,\cdot)\)
\(\chi_{33813}(13958,\cdot)\)
\(\chi_{33813}(14264,\cdot)\)
\(\chi_{33813}(15947,\cdot)\)
\(\chi_{33813}(16253,\cdot)\)
\(\chi_{33813}(17936,\cdot)\)
\(\chi_{33813}(18242,\cdot)\)
\(\chi_{33813}(19925,\cdot)\)
\(\chi_{33813}(21914,\cdot)\)
\(\chi_{33813}(22220,\cdot)\)
\(\chi_{33813}(23903,\cdot)\)
\(\chi_{33813}(24209,\cdot)\)
\(\chi_{33813}(25892,\cdot)\)
\(\chi_{33813}(26198,\cdot)\)
\(\chi_{33813}(27881,\cdot)\)
\(\chi_{33813}(28187,\cdot)\)
\(\chi_{33813}(29870,\cdot)\)
\(\chi_{33813}(30176,\cdot)\)
...
![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 };
\((26300,2602,9829)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{17}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 33813 }(25892, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
