from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(539, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([34,21]))
pari: [g,chi] = znchar(Mod(186,539))
Basic properties
Modulus: | \(539\) | |
Conductor: | \(539\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | 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 539.x
\(\chi_{539}(32,\cdot)\) \(\chi_{539}(65,\cdot)\) \(\chi_{539}(109,\cdot)\) \(\chi_{539}(142,\cdot)\) \(\chi_{539}(186,\cdot)\) \(\chi_{539}(219,\cdot)\) \(\chi_{539}(296,\cdot)\) \(\chi_{539}(340,\cdot)\) \(\chi_{539}(417,\cdot)\) \(\chi_{539}(450,\cdot)\) \(\chi_{539}(494,\cdot)\) \(\chi_{539}(527,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((199,442)\) → \((e\left(\frac{17}{21}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
\( \chi_{ 539 }(186, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) |
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)