from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(850, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([12,3]))
pari: [g,chi] = znchar(Mod(843,850))
Basic properties
| Modulus: | \(850\) | |
| Conductor: | \(85\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{85}(78,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 850.v
\(\chi_{850}(107,\cdot)\) \(\chi_{850}(143,\cdot)\) \(\chi_{850}(193,\cdot)\) \(\chi_{850}(207,\cdot)\) \(\chi_{850}(507,\cdot)\) \(\chi_{850}(607,\cdot)\) \(\chi_{850}(793,\cdot)\) \(\chi_{850}(843,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.698833752810013621337890625.1 |
Values on generators
\((477,751)\) → \((-i,e\left(\frac{3}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 850 }(843, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) |
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)