![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(25425, base_ring=CyclotomicField(140))
M = H._module
chi = DirichletCharacter(H, M([70,7,15]))
![Copy content]()
gp:[g,chi] = znchar(Mod(8477, 25425))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("25425.8477");
| Modulus: | \(25425\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(8475\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(140\) |
![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 \(\chi_{8475}(2,\cdot)\) |
![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);
|
\(\chi_{25425}(8,\cdot)\)
\(\chi_{25425}(53,\cdot)\)
\(\chi_{25425}(512,\cdot)\)
\(\chi_{25425}(1412,\cdot)\)
\(\chi_{25425}(1727,\cdot)\)
\(\chi_{25425}(2258,\cdot)\)
\(\chi_{25425}(3392,\cdot)\)
\(\chi_{25425}(3923,\cdot)\)
\(\chi_{25425}(4238,\cdot)\)
\(\chi_{25425}(5138,\cdot)\)
\(\chi_{25425}(5597,\cdot)\)
\(\chi_{25425}(5642,\cdot)\)
\(\chi_{25425}(6497,\cdot)\)
\(\chi_{25425}(6812,\cdot)\)
\(\chi_{25425}(6992,\cdot)\)
\(\chi_{25425}(8477,\cdot)\)
\(\chi_{25425}(8828,\cdot)\)
\(\chi_{25425}(9008,\cdot)\)
\(\chi_{25425}(9323,\cdot)\)
\(\chi_{25425}(10178,\cdot)\)
\(\chi_{25425}(10223,\cdot)\)
\(\chi_{25425}(10727,\cdot)\)
\(\chi_{25425}(11897,\cdot)\)
\(\chi_{25425}(12077,\cdot)\)
\(\chi_{25425}(12428,\cdot)\)
\(\chi_{25425}(13562,\cdot)\)
\(\chi_{25425}(13913,\cdot)\)
\(\chi_{25425}(14408,\cdot)\)
\(\chi_{25425}(15263,\cdot)\)
\(\chi_{25425}(15308,\cdot)\)
...
![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 };
\((22601,3052,24976)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{3}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 25425 }(8477, a) \) |
\(1\) | \(1\) | \(e\left(\frac{117}{140}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{71}{140}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{43}{140}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{71}{140}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
