from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(776, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,24,41]))
pari: [g,chi] = znchar(Mod(483,776))
Basic properties
| Modulus: | \(776\) | |
| Conductor: | \(776\) | 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 776.bp
\(\chi_{776}(3,\cdot)\) \(\chi_{776}(11,\cdot)\) \(\chi_{776}(99,\cdot)\) \(\chi_{776}(163,\cdot)\) \(\chi_{776}(219,\cdot)\) \(\chi_{776}(243,\cdot)\) \(\chi_{776}(259,\cdot)\) \(\chi_{776}(323,\cdot)\) \(\chi_{776}(339,\cdot)\) \(\chi_{776}(363,\cdot)\) \(\chi_{776}(419,\cdot)\) \(\chi_{776}(483,\cdot)\) \(\chi_{776}(571,\cdot)\) \(\chi_{776}(579,\cdot)\) \(\chi_{776}(635,\cdot)\) \(\chi_{776}(723,\cdot)\)
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
\((583,389,393)\) → \((-1,-1,e\left(\frac{41}{48}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 776 }(483, a) \) | \(-1\) | \(1\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{37}{48}\right)\) |
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)