![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,1,10]))
![Copy content]()
pari:[g,chi] = znchar(Mod(2027,5200))
| Modulus: | \(5200\) | |
| Conductor: | \(5200\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(20\) |
![Copy content]()
sage:chi.multiplicative_order()
![Copy content]()
pari:charorder(g,chi)
|
| Real: | no |
| Primitive: | yes |
![Copy content]()
sage:chi.is_primitive()
![Copy content]()
pari:#znconreyconductor(g,chi)==1
|
| Minimal: | yes |
| Parity: | even |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{5200}(883,\cdot)\)
\(\chi_{5200}(987,\cdot)\)
\(\chi_{5200}(1923,\cdot)\)
\(\chi_{5200}(2027,\cdot)\)
\(\chi_{5200}(2963,\cdot)\)
\(\chi_{5200}(3067,\cdot)\)
\(\chi_{5200}(4003,\cdot)\)
\(\chi_{5200}(5147,\cdot)\)
![Copy content]()
sage:chi.galois_orbit()
![Copy content]()
pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((1951,1301,4577,1601)\) → \((-1,i,e\left(\frac{1}{20}\right),-1)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 5200 }(2027, a) \) |
\(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
