from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(772, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,35]))
pari: [g,chi] = znchar(Mod(543,772))
Basic properties
| Modulus: | \(772\) | |
| Conductor: | \(772\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 772.u
\(\chi_{772}(59,\cdot)\) \(\chi_{772}(131,\cdot)\) \(\chi_{772}(147,\cdot)\) \(\chi_{772}(239,\cdot)\) \(\chi_{772}(255,\cdot)\) \(\chi_{772}(327,\cdot)\) \(\chi_{772}(407,\cdot)\) \(\chi_{772}(531,\cdot)\) \(\chi_{772}(543,\cdot)\) \(\chi_{772}(551,\cdot)\) \(\chi_{772}(575,\cdot)\) \(\chi_{772}(583,\cdot)\) \(\chi_{772}(607,\cdot)\) \(\chi_{772}(615,\cdot)\) \(\chi_{772}(627,\cdot)\) \(\chi_{772}(751,\cdot)\)
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
\((387,5)\) → \((-1,e\left(\frac{35}{48}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 772 }(543, a) \) | \(-1\) | \(1\) | \(-i\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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)