![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91809, base_ring=CyclotomicField(300))
M = H._module
chi = DirichletCharacter(H, M([100,237]))
![Copy content]()
gp:[g,chi] = znchar(Mod(22666, 91809))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91809.22666");
| Modulus: | \(91809\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(909\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(300\) |
![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 \(\chi_{909}(850,\cdot)\) |
![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);
|
\(\chi_{91809}(268,\cdot)\)
\(\chi_{91809}(1744,\cdot)\)
\(\chi_{91809}(1816,\cdot)\)
\(\chi_{91809}(2398,\cdot)\)
\(\chi_{91809}(2497,\cdot)\)
\(\chi_{91809}(2977,\cdot)\)
\(\chi_{91809}(4732,\cdot)\)
\(\chi_{91809}(9454,\cdot)\)
\(\chi_{91809}(10735,\cdot)\)
\(\chi_{91809}(10948,\cdot)\)
\(\chi_{91809}(15046,\cdot)\)
\(\chi_{91809}(17905,\cdot)\)
\(\chi_{91809}(18004,\cdot)\)
\(\chi_{91809}(20650,\cdot)\)
\(\chi_{91809}(20743,\cdot)\)
\(\chi_{91809}(20794,\cdot)\)
\(\chi_{91809}(20857,\cdot)\)
\(\chi_{91809}(20941,\cdot)\)
\(\chi_{91809}(21019,\cdot)\)
\(\chi_{91809}(22666,\cdot)\)
\(\chi_{91809}(23515,\cdot)\)
\(\chi_{91809}(24853,\cdot)\)
\(\chi_{91809}(26152,\cdot)\)
\(\chi_{91809}(27490,\cdot)\)
\(\chi_{91809}(28339,\cdot)\)
\(\chi_{91809}(29986,\cdot)\)
\(\chi_{91809}(30064,\cdot)\)
\(\chi_{91809}(30148,\cdot)\)
\(\chi_{91809}(30211,\cdot)\)
\(\chi_{91809}(30262,\cdot)\)
...
![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 };
\((71408,20404)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{79}{100}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 91809 }(22666, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{37}{300}\right)\) | \(e\left(\frac{37}{150}\right)\) | \(e\left(\frac{47}{75}\right)\) | \(e\left(\frac{133}{300}\right)\) | \(e\left(\frac{37}{100}\right)\) | \(-i\) | \(e\left(\frac{181}{300}\right)\) | \(e\left(\frac{121}{150}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{37}{75}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
