![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5616, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,20,15]))
![Copy content]()
pari:[g,chi] = znchar(Mod(1267,5616))
| Modulus: | \(5616\) | |
| Conductor: | \(5616\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(36\) |
![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)
|
\(\chi_{5616}(67,\cdot)\)
\(\chi_{5616}(427,\cdot)\)
\(\chi_{5616}(475,\cdot)\)
\(\chi_{5616}(1267,\cdot)\)
\(\chi_{5616}(1939,\cdot)\)
\(\chi_{5616}(2299,\cdot)\)
\(\chi_{5616}(2347,\cdot)\)
\(\chi_{5616}(3139,\cdot)\)
\(\chi_{5616}(3811,\cdot)\)
\(\chi_{5616}(4171,\cdot)\)
\(\chi_{5616}(4219,\cdot)\)
\(\chi_{5616}(5011,\cdot)\)
![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 ]
\((703,4213,2081,3889)\) → \((-1,-i,e\left(\frac{5}{9}\right),e\left(\frac{5}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 5616 }(1267, a) \) |
\(1\) | \(1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-1\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(-i\) |
![Copy content]()
sage:chi.jacobi_sum(n)
