from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(86, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([3]))
pari: [g,chi] = znchar(Mod(75,86))
Basic properties
Modulus: | \(86\) | |
Conductor: | \(43\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{43}(32,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 86.f
\(\chi_{86}(27,\cdot)\) \(\chi_{86}(39,\cdot)\) \(\chi_{86}(45,\cdot)\) \(\chi_{86}(51,\cdot)\) \(\chi_{86}(65,\cdot)\) \(\chi_{86}(75,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
Fixed field: | Number field defined by a degree 14 polynomial |
Values on generators
\(3\) → \(e\left(\frac{3}{14}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 86 }(75, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) |
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)