![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6195, base_ring=CyclotomicField(348))
M = H._module
chi = DirichletCharacter(H, M([174,261,58,48]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1613, 6195))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6195.1613");
| Modulus: | \(6195\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(6195\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(348\) |
![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);
|
\(\chi_{6195}(17,\cdot)\)
\(\chi_{6195}(68,\cdot)\)
\(\chi_{6195}(122,\cdot)\)
\(\chi_{6195}(143,\cdot)\)
\(\chi_{6195}(248,\cdot)\)
\(\chi_{6195}(257,\cdot)\)
\(\chi_{6195}(383,\cdot)\)
\(\chi_{6195}(458,\cdot)\)
\(\chi_{6195}(488,\cdot)\)
\(\chi_{6195}(572,\cdot)\)
\(\chi_{6195}(593,\cdot)\)
\(\chi_{6195}(647,\cdot)\)
\(\chi_{6195}(668,\cdot)\)
\(\chi_{6195}(677,\cdot)\)
\(\chi_{6195}(698,\cdot)\)
\(\chi_{6195}(782,\cdot)\)
\(\chi_{6195}(803,\cdot)\)
\(\chi_{6195}(992,\cdot)\)
\(\chi_{6195}(1067,\cdot)\)
\(\chi_{6195}(1088,\cdot)\)
\(\chi_{6195}(1097,\cdot)\)
\(\chi_{6195}(1172,\cdot)\)
\(\chi_{6195}(1202,\cdot)\)
\(\chi_{6195}(1307,\cdot)\)
\(\chi_{6195}(1382,\cdot)\)
\(\chi_{6195}(1403,\cdot)\)
\(\chi_{6195}(1433,\cdot)\)
\(\chi_{6195}(1487,\cdot)\)
\(\chi_{6195}(1538,\cdot)\)
\(\chi_{6195}(1613,\cdot)\)
...
![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 };
\((2066,4957,2656,946)\) → \((-1,-i,e\left(\frac{1}{6}\right),e\left(\frac{4}{29}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 6195 }(1613, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{251}{348}\right)\) | \(e\left(\frac{77}{174}\right)\) | \(e\left(\frac{19}{116}\right)\) | \(e\left(\frac{107}{174}\right)\) | \(e\left(\frac{111}{116}\right)\) | \(e\left(\frac{77}{87}\right)\) | \(e\left(\frac{325}{348}\right)\) | \(e\left(\frac{50}{87}\right)\) | \(e\left(\frac{39}{116}\right)\) | \(e\left(\frac{53}{348}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
