from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(640, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,31,16]))
pari: [g,chi] = znchar(Mod(179,640))
Basic properties
Modulus: | \(640\) | |
Conductor: | \(640\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 640.bn
\(\chi_{640}(19,\cdot)\) \(\chi_{640}(59,\cdot)\) \(\chi_{640}(99,\cdot)\) \(\chi_{640}(139,\cdot)\) \(\chi_{640}(179,\cdot)\) \(\chi_{640}(219,\cdot)\) \(\chi_{640}(259,\cdot)\) \(\chi_{640}(299,\cdot)\) \(\chi_{640}(339,\cdot)\) \(\chi_{640}(379,\cdot)\) \(\chi_{640}(419,\cdot)\) \(\chi_{640}(459,\cdot)\) \(\chi_{640}(499,\cdot)\) \(\chi_{640}(539,\cdot)\) \(\chi_{640}(579,\cdot)\) \(\chi_{640}(619,\cdot)\)
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: | 32.0.478904856520590268236983445984471619880855975682375680000000000000000.1 |
Values on generators
\((511,261,257)\) → \((-1,e\left(\frac{31}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 640 }(179, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) |
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)