![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([77,28,121]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1223, 2001))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2001.1223");
| Modulus: | \(2001\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2001\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{2001}(35,\cdot)\)
\(\chi_{2001}(62,\cdot)\)
\(\chi_{2001}(71,\cdot)\)
\(\chi_{2001}(167,\cdot)\)
\(\chi_{2001}(179,\cdot)\)
\(\chi_{2001}(209,\cdot)\)
\(\chi_{2001}(236,\cdot)\)
\(\chi_{2001}(266,\cdot)\)
\(\chi_{2001}(353,\cdot)\)
\(\chi_{2001}(386,\cdot)\)
\(\chi_{2001}(440,\cdot)\)
\(\chi_{2001}(473,\cdot)\)
\(\chi_{2001}(515,\cdot)\)
\(\chi_{2001}(560,\cdot)\)
\(\chi_{2001}(584,\cdot)\)
\(\chi_{2001}(593,\cdot)\)
\(\chi_{2001}(602,\cdot)\)
\(\chi_{2001}(614,\cdot)\)
\(\chi_{2001}(647,\cdot)\)
\(\chi_{2001}(671,\cdot)\)
\(\chi_{2001}(680,\cdot)\)
\(\chi_{2001}(731,\cdot)\)
\(\chi_{2001}(767,\cdot)\)
\(\chi_{2001}(788,\cdot)\)
\(\chi_{2001}(818,\cdot)\)
\(\chi_{2001}(821,\cdot)\)
\(\chi_{2001}(854,\cdot)\)
\(\chi_{2001}(863,\cdot)\)
\(\chi_{2001}(905,\cdot)\)
\(\chi_{2001}(932,\cdot)\)
...
![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 };
\((668,1132,553)\) → \((-1,e\left(\frac{2}{11}\right),e\left(\frac{11}{14}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2001 }(1223, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{149}{154}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{95}{154}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
