![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1300, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,1,15]))
![Copy content]()
pari:[g,chi] = znchar(Mod(1227,1300))
| Modulus: | \(1300\) | |
| Conductor: | \(1300\) |
![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: | odd |
![Copy content]()
sage:chi.is_odd()
![Copy content]()
pari:zncharisodd(g,chi)
|
\(\chi_{1300}(187,\cdot)\)
\(\chi_{1300}(203,\cdot)\)
\(\chi_{1300}(447,\cdot)\)
\(\chi_{1300}(463,\cdot)\)
\(\chi_{1300}(723,\cdot)\)
\(\chi_{1300}(967,\cdot)\)
\(\chi_{1300}(983,\cdot)\)
\(\chi_{1300}(1227,\cdot)\)
![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 ]
\((651,677,301)\) → \((-1,e\left(\frac{1}{20}\right),-i)\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 1300 }(1227, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
![Copy content]()
sage:chi.jacobi_sum(n)
