from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(76, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,7]))
pari: [g,chi] = znchar(Mod(71,76))
Basic properties
Modulus: | \(76\) | |
Conductor: | \(76\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(18\) | 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 76.k
\(\chi_{76}(3,\cdot)\) \(\chi_{76}(15,\cdot)\) \(\chi_{76}(51,\cdot)\) \(\chi_{76}(59,\cdot)\) \(\chi_{76}(67,\cdot)\) \(\chi_{76}(71,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Fixed field: | \(\Q(\zeta_{76})^+\) |
Values on generators
\((39,21)\) → \((-1,e\left(\frac{7}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
\( \chi_{ 76 }(71, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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)