![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6099, base_ring=CyclotomicField(106))
M = H._module
chi = DirichletCharacter(H, M([53,53,74]))
![Copy content]()
gp:[g,chi] = znchar(Mod(797, 6099))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6099.797");
| Modulus: | \(6099\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(6099\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(106\) |
![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);
|
\(\chi_{6099}(56,\cdot)\)
\(\chi_{6099}(227,\cdot)\)
\(\chi_{6099}(455,\cdot)\)
\(\chi_{6099}(569,\cdot)\)
\(\chi_{6099}(683,\cdot)\)
\(\chi_{6099}(797,\cdot)\)
\(\chi_{6099}(854,\cdot)\)
\(\chi_{6099}(1025,\cdot)\)
\(\chi_{6099}(1082,\cdot)\)
\(\chi_{6099}(1139,\cdot)\)
\(\chi_{6099}(1196,\cdot)\)
\(\chi_{6099}(1253,\cdot)\)
\(\chi_{6099}(1367,\cdot)\)
\(\chi_{6099}(1424,\cdot)\)
\(\chi_{6099}(1481,\cdot)\)
\(\chi_{6099}(1538,\cdot)\)
\(\chi_{6099}(1652,\cdot)\)
\(\chi_{6099}(1823,\cdot)\)
\(\chi_{6099}(1880,\cdot)\)
\(\chi_{6099}(1937,\cdot)\)
\(\chi_{6099}(2108,\cdot)\)
\(\chi_{6099}(2165,\cdot)\)
\(\chi_{6099}(2336,\cdot)\)
\(\chi_{6099}(2393,\cdot)\)
\(\chi_{6099}(2621,\cdot)\)
\(\chi_{6099}(2678,\cdot)\)
\(\chi_{6099}(2792,\cdot)\)
\(\chi_{6099}(3077,\cdot)\)
\(\chi_{6099}(3419,\cdot)\)
\(\chi_{6099}(3476,\cdot)\)
...
![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 };
\((4067,2890,5245)\) → \((-1,-1,e\left(\frac{37}{53}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 6099 }(797, a) \) |
\(1\) | \(1\) | \(e\left(\frac{37}{53}\right)\) | \(e\left(\frac{21}{53}\right)\) | \(e\left(\frac{33}{106}\right)\) | \(e\left(\frac{1}{53}\right)\) | \(e\left(\frac{5}{53}\right)\) | \(e\left(\frac{1}{106}\right)\) | \(e\left(\frac{91}{106}\right)\) | \(e\left(\frac{29}{106}\right)\) | \(e\left(\frac{38}{53}\right)\) | \(e\left(\frac{42}{53}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
