from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(580, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,7,10]))
pari: [g,chi] = znchar(Mod(139,580))
Basic properties
Modulus: | \(580\) | |
Conductor: | \(580\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | 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 580.v
\(\chi_{580}(139,\cdot)\) \(\chi_{580}(199,\cdot)\) \(\chi_{580}(219,\cdot)\) \(\chi_{580}(239,\cdot)\) \(\chi_{580}(339,\cdot)\) \(\chi_{580}(459,\cdot)\)
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
\((291,117,321)\) → \((-1,-1,e\left(\frac{5}{7}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 580 }(139, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(-1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) |
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)