![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6175, base_ring=CyclotomicField(180))
M = H._module
chi = DirichletCharacter(H, M([99,45,50]))
![Copy content]()
gp:[g,chi] = znchar(Mod(2673, 6175))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6175.2673");
| Modulus: | \(6175\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(6175\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(180\) |
![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);
|
\(\chi_{6175}(203,\cdot)\)
\(\chi_{6175}(317,\cdot)\)
\(\chi_{6175}(333,\cdot)\)
\(\chi_{6175}(447,\cdot)\)
\(\chi_{6175}(528,\cdot)\)
\(\chi_{6175}(642,\cdot)\)
\(\chi_{6175}(983,\cdot)\)
\(\chi_{6175}(1048,\cdot)\)
\(\chi_{6175}(1097,\cdot)\)
\(\chi_{6175}(1162,\cdot)\)
\(\chi_{6175}(1438,\cdot)\)
\(\chi_{6175}(1503,\cdot)\)
\(\chi_{6175}(1552,\cdot)\)
\(\chi_{6175}(1617,\cdot)\)
\(\chi_{6175}(1763,\cdot)\)
\(\chi_{6175}(1877,\cdot)\)
\(\chi_{6175}(2283,\cdot)\)
\(\chi_{6175}(2397,\cdot)\)
\(\chi_{6175}(2673,\cdot)\)
\(\chi_{6175}(2738,\cdot)\)
\(\chi_{6175}(2787,\cdot)\)
\(\chi_{6175}(2803,\cdot)\)
\(\chi_{6175}(2852,\cdot)\)
\(\chi_{6175}(2917,\cdot)\)
\(\chi_{6175}(2998,\cdot)\)
\(\chi_{6175}(3112,\cdot)\)
\(\chi_{6175}(3453,\cdot)\)
\(\chi_{6175}(3567,\cdot)\)
\(\chi_{6175}(3908,\cdot)\)
\(\chi_{6175}(3973,\cdot)\)
...
![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 };
\((1977,951,3251)\) → \((e\left(\frac{11}{20}\right),i,e\left(\frac{5}{18}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
| \( \chi_{ 6175 }(2673, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{7}{90}\right)\) | \(e\left(\frac{83}{180}\right)\) | \(e\left(\frac{7}{45}\right)\) | \(e\left(\frac{97}{180}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{83}{90}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{45}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
