![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(891, base_ring=CyclotomicField(270))
M = H._module
chi = DirichletCharacter(H, M([245,162]))
![Copy content]()
gp:[g,chi] = znchar(Mod(119, 891))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("891.119");
| Modulus: | \(891\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(891\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(270\) |
![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);
|
\(\chi_{891}(5,\cdot)\)
\(\chi_{891}(14,\cdot)\)
\(\chi_{891}(20,\cdot)\)
\(\chi_{891}(38,\cdot)\)
\(\chi_{891}(47,\cdot)\)
\(\chi_{891}(59,\cdot)\)
\(\chi_{891}(86,\cdot)\)
\(\chi_{891}(92,\cdot)\)
\(\chi_{891}(104,\cdot)\)
\(\chi_{891}(113,\cdot)\)
\(\chi_{891}(119,\cdot)\)
\(\chi_{891}(137,\cdot)\)
\(\chi_{891}(146,\cdot)\)
\(\chi_{891}(158,\cdot)\)
\(\chi_{891}(185,\cdot)\)
\(\chi_{891}(191,\cdot)\)
\(\chi_{891}(203,\cdot)\)
\(\chi_{891}(212,\cdot)\)
\(\chi_{891}(218,\cdot)\)
\(\chi_{891}(236,\cdot)\)
\(\chi_{891}(245,\cdot)\)
\(\chi_{891}(257,\cdot)\)
\(\chi_{891}(284,\cdot)\)
\(\chi_{891}(290,\cdot)\)
\(\chi_{891}(302,\cdot)\)
\(\chi_{891}(311,\cdot)\)
\(\chi_{891}(317,\cdot)\)
\(\chi_{891}(335,\cdot)\)
\(\chi_{891}(344,\cdot)\)
\(\chi_{891}(356,\cdot)\)
...
![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 };
\((650,244)\) → \((e\left(\frac{49}{54}\right),e\left(\frac{3}{5}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 891 }(119, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{137}{270}\right)\) | \(e\left(\frac{2}{135}\right)\) | \(e\left(\frac{73}{270}\right)\) | \(e\left(\frac{97}{135}\right)\) | \(e\left(\frac{47}{90}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{116}{135}\right)\) | \(e\left(\frac{61}{270}\right)\) | \(e\left(\frac{4}{135}\right)\) | \(e\left(\frac{31}{90}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
gp:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
