from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(714, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,32,21]))
pari: [g,chi] = znchar(Mod(11,714))
Basic properties
| Modulus: | \(714\) | |
| Conductor: | \(357\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{357}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 714.bk
\(\chi_{714}(11,\cdot)\) \(\chi_{714}(23,\cdot)\) \(\chi_{714}(65,\cdot)\) \(\chi_{714}(95,\cdot)\) \(\chi_{714}(107,\cdot)\) \(\chi_{714}(233,\cdot)\) \(\chi_{714}(275,\cdot)\) \(\chi_{714}(317,\cdot)\) \(\chi_{714}(347,\cdot)\) \(\chi_{714}(401,\cdot)\) \(\chi_{714}(431,\cdot)\) \(\chi_{714}(473,\cdot)\) \(\chi_{714}(515,\cdot)\) \(\chi_{714}(641,\cdot)\) \(\chi_{714}(653,\cdot)\) \(\chi_{714}(683,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((239,409,547)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{7}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 714 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{5}{16}\right)\) |
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)