![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(257725, base_ring=CyclotomicField(780))
M = H._module
chi = DirichletCharacter(H, M([507,170,299]))
![Copy content]()
gp:[g,chi] = znchar(Mod(19042, 257725))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("257725.19042");
| Modulus: | \(257725\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(257725\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(780\) |
![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);
|
\(\chi_{257725}(322,\cdot)\)
\(\chi_{257725}(7303,\cdot)\)
\(\chi_{257725}(7388,\cdot)\)
\(\chi_{257725}(8298,\cdot)\)
\(\chi_{257725}(8408,\cdot)\)
\(\chi_{257725}(8558,\cdot)\)
\(\chi_{257725}(10462,\cdot)\)
\(\chi_{257725}(10527,\cdot)\)
\(\chi_{257725}(14317,\cdot)\)
\(\chi_{257725}(14577,\cdot)\)
\(\chi_{257725}(15487,\cdot)\)
\(\chi_{257725}(16988,\cdot)\)
\(\chi_{257725}(18672,\cdot)\)
\(\chi_{257725}(19042,\cdot)\)
\(\chi_{257725}(20147,\cdot)\)
\(\chi_{257725}(24028,\cdot)\)
\(\chi_{257725}(27128,\cdot)\)
\(\chi_{257725}(27213,\cdot)\)
\(\chi_{257725}(28123,\cdot)\)
\(\chi_{257725}(28233,\cdot)\)
\(\chi_{257725}(28383,\cdot)\)
\(\chi_{257725}(30287,\cdot)\)
\(\chi_{257725}(30352,\cdot)\)
\(\chi_{257725}(34142,\cdot)\)
\(\chi_{257725}(34402,\cdot)\)
\(\chi_{257725}(35312,\cdot)\)
\(\chi_{257725}(36748,\cdot)\)
\(\chi_{257725}(36813,\cdot)\)
\(\chi_{257725}(38497,\cdot)\)
\(\chi_{257725}(38867,\cdot)\)
...
![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 };
\((144327,193676,177451)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{17}{78}\right),e\left(\frac{23}{60}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 257725 }(19042, a) \) |
\(1\) | \(1\) | \(e\left(\frac{49}{195}\right)\) | \(e\left(\frac{683}{780}\right)\) | \(e\left(\frac{98}{195}\right)\) | \(e\left(\frac{33}{260}\right)\) | \(e\left(\frac{23}{65}\right)\) | \(e\left(\frac{49}{65}\right)\) | \(e\left(\frac{293}{390}\right)\) | \(e\left(\frac{467}{780}\right)\) | \(e\left(\frac{59}{156}\right)\) | \(e\left(\frac{118}{195}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
