from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([8,7]))
pari: [g,chi] = znchar(Mod(11,91))
Basic properties
Modulus: | \(91\) | |
Conductor: | \(91\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 91.bd
\(\chi_{91}(11,\cdot)\) \(\chi_{91}(58,\cdot)\) \(\chi_{91}(67,\cdot)\) \(\chi_{91}(72,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
Fixed field: | 12.0.10331448031704891637.2 |
Values on generators
\((66,15)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{12}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 91 }(11, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(-i\) | \(1\) | \(-1\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) |
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)