![Copy content]()
sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(143, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([18,35]))
![Copy content]()
pari:[g,chi] = znchar(Mod(63,143))
| Modulus: | \(143\) | |
| Conductor: | \(143\) |
![Copy content]()
sage:chi.conductor()
![Copy content]()
pari:znconreyconductor(g,chi)
|
| Order: | \(60\) |
![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)
|
\(\chi_{143}(2,\cdot)\)
\(\chi_{143}(6,\cdot)\)
\(\chi_{143}(7,\cdot)\)
\(\chi_{143}(19,\cdot)\)
\(\chi_{143}(24,\cdot)\)
\(\chi_{143}(28,\cdot)\)
\(\chi_{143}(41,\cdot)\)
\(\chi_{143}(46,\cdot)\)
\(\chi_{143}(50,\cdot)\)
\(\chi_{143}(63,\cdot)\)
\(\chi_{143}(72,\cdot)\)
\(\chi_{143}(84,\cdot)\)
\(\chi_{143}(85,\cdot)\)
\(\chi_{143}(106,\cdot)\)
\(\chi_{143}(123,\cdot)\)
\(\chi_{143}(128,\cdot)\)
![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 ]
\((79,67)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{7}{12}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 143 }(63, a) \) |
\(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) |
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.gauss_sum(a)
![Copy content]()
pari:znchargauss(g,chi,a)
![Copy content]()
sage:chi.jacobi_sum(n)
![Copy content]()
sage:chi.kloosterman_sum(a,b)
