![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1453, base_ring=CyclotomicField(726))
M = H._module
chi = DirichletCharacter(H, M([104]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1017, 1453))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1453.1017");
| Modulus: | \(1453\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(1453\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(363\) |
![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);
|
\(\chi_{1453}(3,\cdot)\)
\(\chi_{1453}(9,\cdot)\)
\(\chi_{1453}(13,\cdot)\)
\(\chi_{1453}(16,\cdot)\)
\(\chi_{1453}(21,\cdot)\)
\(\chi_{1453}(23,\cdot)\)
\(\chi_{1453}(37,\cdot)\)
\(\chi_{1453}(40,\cdot)\)
\(\chi_{1453}(77,\cdot)\)
\(\chi_{1453}(81,\cdot)\)
\(\chi_{1453}(91,\cdot)\)
\(\chi_{1453}(97,\cdot)\)
\(\chi_{1453}(99,\cdot)\)
\(\chi_{1453}(100,\cdot)\)
\(\chi_{1453}(107,\cdot)\)
\(\chi_{1453}(109,\cdot)\)
\(\chi_{1453}(112,\cdot)\)
\(\chi_{1453}(117,\cdot)\)
\(\chi_{1453}(118,\cdot)\)
\(\chi_{1453}(134,\cdot)\)
\(\chi_{1453}(139,\cdot)\)
\(\chi_{1453}(142,\cdot)\)
\(\chi_{1453}(143,\cdot)\)
\(\chi_{1453}(147,\cdot)\)
\(\chi_{1453}(151,\cdot)\)
\(\chi_{1453}(161,\cdot)\)
\(\chi_{1453}(169,\cdot)\)
\(\chi_{1453}(174,\cdot)\)
\(\chi_{1453}(176,\cdot)\)
\(\chi_{1453}(181,\cdot)\)
...
![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 };
\(2\) → \(e\left(\frac{52}{363}\right)\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1453 }(1017, a) \) |
\(1\) | \(1\) | \(e\left(\frac{52}{363}\right)\) | \(e\left(\frac{116}{363}\right)\) | \(e\left(\frac{104}{363}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{56}{121}\right)\) | \(e\left(\frac{40}{121}\right)\) | \(e\left(\frac{52}{121}\right)\) | \(e\left(\frac{232}{363}\right)\) | \(e\left(\frac{74}{363}\right)\) | \(e\left(\frac{23}{33}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
