![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([77,110,105]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1583, 3381))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3381.1583");
| Modulus: | \(3381\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(3381\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(154\) |
![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);
|
\(\chi_{3381}(113,\cdot)\)
\(\chi_{3381}(134,\cdot)\)
\(\chi_{3381}(155,\cdot)\)
\(\chi_{3381}(176,\cdot)\)
\(\chi_{3381}(218,\cdot)\)
\(\chi_{3381}(260,\cdot)\)
\(\chi_{3381}(281,\cdot)\)
\(\chi_{3381}(365,\cdot)\)
\(\chi_{3381}(428,\cdot)\)
\(\chi_{3381}(470,\cdot)\)
\(\chi_{3381}(596,\cdot)\)
\(\chi_{3381}(617,\cdot)\)
\(\chi_{3381}(659,\cdot)\)
\(\chi_{3381}(701,\cdot)\)
\(\chi_{3381}(743,\cdot)\)
\(\chi_{3381}(764,\cdot)\)
\(\chi_{3381}(848,\cdot)\)
\(\chi_{3381}(911,\cdot)\)
\(\chi_{3381}(953,\cdot)\)
\(\chi_{3381}(1100,\cdot)\)
\(\chi_{3381}(1121,\cdot)\)
\(\chi_{3381}(1142,\cdot)\)
\(\chi_{3381}(1184,\cdot)\)
\(\chi_{3381}(1247,\cdot)\)
\(\chi_{3381}(1331,\cdot)\)
\(\chi_{3381}(1394,\cdot)\)
\(\chi_{3381}(1436,\cdot)\)
\(\chi_{3381}(1562,\cdot)\)
\(\chi_{3381}(1583,\cdot)\)
\(\chi_{3381}(1604,\cdot)\)
...
![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 };
\((2255,346,442)\) → \((-1,e\left(\frac{5}{7}\right),e\left(\frac{15}{22}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3381 }(1583, a) \) |
\(1\) | \(1\) | \(e\left(\frac{67}{154}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{51}{154}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{5}{22}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
