from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([12,4,15]))
pari: [g,chi] = znchar(Mod(59,714))
Basic properties
Modulus: | \(714\) | |
Conductor: | \(357\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{357}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 714.bj
\(\chi_{714}(59,\cdot)\) \(\chi_{714}(185,\cdot)\) \(\chi_{714}(257,\cdot)\) \(\chi_{714}(383,\cdot)\) \(\chi_{714}(467,\cdot)\) \(\chi_{714}(563,\cdot)\) \(\chi_{714}(593,\cdot)\) \(\chi_{714}(689,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
Fixed field: | Number field defined by a degree 24 polynomial |
Values on generators
\((239,409,547)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{5}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 714 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) |
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)