from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(561, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,4,35]))
pari: [g,chi] = znchar(Mod(2,561))
Basic properties
Modulus: | \(561\) | |
Conductor: | \(561\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 561.bg
\(\chi_{561}(2,\cdot)\) \(\chi_{561}(8,\cdot)\) \(\chi_{561}(83,\cdot)\) \(\chi_{561}(128,\cdot)\) \(\chi_{561}(134,\cdot)\) \(\chi_{561}(161,\cdot)\) \(\chi_{561}(206,\cdot)\) \(\chi_{561}(281,\cdot)\) \(\chi_{561}(314,\cdot)\) \(\chi_{561}(332,\cdot)\) \(\chi_{561}(338,\cdot)\) \(\chi_{561}(359,\cdot)\) \(\chi_{561}(365,\cdot)\) \(\chi_{561}(491,\cdot)\) \(\chi_{561}(512,\cdot)\) \(\chi_{561}(536,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((188,409,496)\) → \((-1,e\left(\frac{1}{10}\right),e\left(\frac{7}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(19\) |
\( \chi_{ 561 }(2, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)