![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22385, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([135,36,20]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1533, 22385))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22385.1533");
| Modulus: | \(22385\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(2035\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(180\) |
![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: | no, induced from \(\chi_{2035}(1533,\cdot)\) |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
gp:#znconreyconductor(g,chi)==1
![Copy content]()
magma:IsPrimitive(chi);
|
| Minimal: | no |
| Parity: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
gp:zncharisodd(g,chi)
![Copy content]()
magma:IsOdd(chi);
|
\(\chi_{22385}(493,\cdot)\)
\(\chi_{22385}(608,\cdot)\)
\(\chi_{22385}(1237,\cdot)\)
\(\chi_{22385}(1533,\cdot)\)
\(\chi_{22385}(3512,\cdot)\)
\(\chi_{22385}(3633,\cdot)\)
\(\chi_{22385}(4123,\cdot)\)
\(\chi_{22385}(4262,\cdot)\)
\(\chi_{22385}(4437,\cdot)\)
\(\chi_{22385}(4558,\cdot)\)
\(\chi_{22385}(4843,\cdot)\)
\(\chi_{22385}(5448,\cdot)\)
\(\chi_{22385}(5472,\cdot)\)
\(\chi_{22385}(5768,\cdot)\)
\(\chi_{22385}(5932,\cdot)\)
\(\chi_{22385}(6077,\cdot)\)
\(\chi_{22385}(6373,\cdot)\)
\(\chi_{22385}(6857,\cdot)\)
\(\chi_{22385}(7027,\cdot)\)
\(\chi_{22385}(7148,\cdot)\)
\(\chi_{22385}(8358,\cdot)\)
\(\chi_{22385}(8618,\cdot)\)
\(\chi_{22385}(8963,\cdot)\)
\(\chi_{22385}(9447,\cdot)\)
\(\chi_{22385}(9562,\cdot)\)
\(\chi_{22385}(10487,\cdot)\)
\(\chi_{22385}(11038,\cdot)\)
\(\chi_{22385}(12587,\cdot)\)
\(\chi_{22385}(13077,\cdot)\)
\(\chi_{22385}(13512,\cdot)\)
...
![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 };
\((13432,19241,12101)\) → \((-i,e\left(\frac{1}{5}\right),e\left(\frac{1}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
| \( \chi_{ 22385 }(1533, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{11}{180}\right)\) | \(e\left(\frac{133}{180}\right)\) | \(e\left(\frac{11}{90}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{127}{180}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{43}{90}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{121}{180}\right)\) | \(e\left(\frac{23}{30}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
