![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32683, base_ring=CyclotomicField(308))
M = H._module
chi = DirichletCharacter(H, M([110,210,99]))
![Copy content]()
gp:[g,chi] = znchar(Mod(2687, 32683))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32683.2687");
| Modulus: | \(32683\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(32683\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(308\) |
![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);
|
\(\chi_{32683}(356,\cdot)\)
\(\chi_{32683}(475,\cdot)\)
\(\chi_{32683}(566,\cdot)\)
\(\chi_{32683}(1203,\cdot)\)
\(\chi_{32683}(1539,\cdot)\)
\(\chi_{32683}(1700,\cdot)\)
\(\chi_{32683}(1896,\cdot)\)
\(\chi_{32683}(2022,\cdot)\)
\(\chi_{32683}(2666,\cdot)\)
\(\chi_{32683}(2687,\cdot)\)
\(\chi_{32683}(3317,\cdot)\)
\(\chi_{32683}(4045,\cdot)\)
\(\chi_{32683}(4108,\cdot)\)
\(\chi_{32683}(4157,\cdot)\)
\(\chi_{32683}(4318,\cdot)\)
\(\chi_{32683}(4381,\cdot)\)
\(\chi_{32683}(4542,\cdot)\)
\(\chi_{32683}(4619,\cdot)\)
\(\chi_{32683}(4759,\cdot)\)
\(\chi_{32683}(4864,\cdot)\)
\(\chi_{32683}(5466,\cdot)\)
\(\chi_{32683}(5508,\cdot)\)
\(\chi_{32683}(6040,\cdot)\)
\(\chi_{32683}(6250,\cdot)\)
\(\chi_{32683}(6887,\cdot)\)
\(\chi_{32683}(6999,\cdot)\)
\(\chi_{32683}(7160,\cdot)\)
\(\chi_{32683}(7601,\cdot)\)
\(\chi_{32683}(8308,\cdot)\)
\(\chi_{32683}(8644,\cdot)\)
...
![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 };
\((24013,18474,7890)\) → \((e\left(\frac{5}{14}\right),e\left(\frac{15}{22}\right),e\left(\frac{9}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 32683 }(2687, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{299}{308}\right)\) | \(e\left(\frac{269}{308}\right)\) | \(e\left(\frac{145}{154}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{281}{308}\right)\) | \(e\left(\frac{115}{154}\right)\) | \(e\left(\frac{25}{308}\right)\) | \(e\left(\frac{141}{308}\right)\) | \(e\left(\frac{251}{308}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
