![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4477, base_ring=CyclotomicField(198))
M = H._module
chi = DirichletCharacter(H, M([81,88]))
![Copy content]()
gp:[g,chi] = znchar(Mod(1748, 4477))
![Copy content]()
magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4477.1748");
| Modulus: | \(4477\) |
![Copy content]()
sage:chi.modulus()
![Copy content]()
gp:g[1][1]
![Copy content]()
magma:Modulus(chi);
|
| Conductor: | \(4477\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
gp:znconreyconductor(g,chi)
![Copy content]()
magma:Conductor(chi);
|
| Order: | \(198\) |
![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);
|
\(\chi_{4477}(164,\cdot)\)
\(\chi_{4477}(197,\cdot)\)
\(\chi_{4477}(219,\cdot)\)
\(\chi_{4477}(329,\cdot)\)
\(\chi_{4477}(340,\cdot)\)
\(\chi_{4477}(527,\cdot)\)
\(\chi_{4477}(571,\cdot)\)
\(\chi_{4477}(626,\cdot)\)
\(\chi_{4477}(736,\cdot)\)
\(\chi_{4477}(747,\cdot)\)
\(\chi_{4477}(934,\cdot)\)
\(\chi_{4477}(978,\cdot)\)
\(\chi_{4477}(1011,\cdot)\)
\(\chi_{4477}(1033,\cdot)\)
\(\chi_{4477}(1143,\cdot)\)
\(\chi_{4477}(1154,\cdot)\)
\(\chi_{4477}(1341,\cdot)\)
\(\chi_{4477}(1385,\cdot)\)
\(\chi_{4477}(1418,\cdot)\)
\(\chi_{4477}(1440,\cdot)\)
\(\chi_{4477}(1550,\cdot)\)
\(\chi_{4477}(1561,\cdot)\)
\(\chi_{4477}(1748,\cdot)\)
\(\chi_{4477}(1792,\cdot)\)
\(\chi_{4477}(1825,\cdot)\)
\(\chi_{4477}(1847,\cdot)\)
\(\chi_{4477}(1957,\cdot)\)
\(\chi_{4477}(1968,\cdot)\)
\(\chi_{4477}(2155,\cdot)\)
\(\chi_{4477}(2199,\cdot)\)
...
![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 };
\((1333,3147)\) → \((e\left(\frac{9}{22}\right),e\left(\frac{4}{9}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 4477 }(1748, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{169}{198}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{70}{99}\right)\) | \(e\left(\frac{49}{99}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{17}{198}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{26}{99}\right)\) |
![Copy content]()
sage:chi(x)
![Copy content]()
gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
![Copy content]()
magma:chi(x)
