from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(257, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([27]))
pari: [g,chi] = znchar(Mod(68,257))
Basic properties
| Modulus: | \(257\) | |
| Conductor: | \(257\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 257.f
\(\chi_{257}(15,\cdot)\) \(\chi_{257}(17,\cdot)\) \(\chi_{257}(30,\cdot)\) \(\chi_{257}(34,\cdot)\) \(\chi_{257}(60,\cdot)\) \(\chi_{257}(68,\cdot)\) \(\chi_{257}(120,\cdot)\) \(\chi_{257}(121,\cdot)\) \(\chi_{257}(136,\cdot)\) \(\chi_{257}(137,\cdot)\) \(\chi_{257}(189,\cdot)\) \(\chi_{257}(197,\cdot)\) \(\chi_{257}(223,\cdot)\) \(\chi_{257}(227,\cdot)\) \(\chi_{257}(240,\cdot)\) \(\chi_{257}(242,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\(3\) → \(e\left(\frac{27}{32}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 257 }(68, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{27}{32}\right)\) | \(1\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(-1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) |
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)