from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(80, base_ring=CyclotomicField(4))
M = H._module
chi = DirichletCharacter(H, M([2,0,1]))
pari: [g,chi] = znchar(Mod(47,80))
Basic properties
Modulus: | \(80\) | |
Conductor: | \(20\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{20}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 80.n
\(\chi_{80}(47,\cdot)\) \(\chi_{80}(63,\cdot)\)
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: | \(\mathbb{Q}(i)\) |
Fixed field: | \(\Q(\zeta_{20})^+\) |
Values on generators
\((31,21,17)\) → \((-1,1,i)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 80 }(47, a) \) | \(1\) | \(1\) | \(i\) | \(-i\) | \(-1\) | \(-1\) | \(-i\) | \(i\) | \(1\) | \(1\) | \(i\) | \(-i\) |
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)