from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,17]))
pari: [g,chi] = znchar(Mod(153,256))
Basic properties
Modulus: | \(256\) | |
Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{128}(69,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 256.k
\(\chi_{256}(9,\cdot)\) \(\chi_{256}(25,\cdot)\) \(\chi_{256}(41,\cdot)\) \(\chi_{256}(57,\cdot)\) \(\chi_{256}(73,\cdot)\) \(\chi_{256}(89,\cdot)\) \(\chi_{256}(105,\cdot)\) \(\chi_{256}(121,\cdot)\) \(\chi_{256}(137,\cdot)\) \(\chi_{256}(153,\cdot)\) \(\chi_{256}(169,\cdot)\) \(\chi_{256}(185,\cdot)\) \(\chi_{256}(201,\cdot)\) \(\chi_{256}(217,\cdot)\) \(\chi_{256}(233,\cdot)\) \(\chi_{256}(249,\cdot)\)
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: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((255,5)\) → \((1,e\left(\frac{17}{32}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 256 }(153, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) |
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)