Properties

Label 256.153
Modulus $256$
Conductor $128$
Order $32$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
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)
 
\( \chi_{ 256 }(153,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 256 }(153,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 256 }(153,·),\chi_{ 256 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 256 }(153,·)) \;\) at \(\; a,b = \) e.g. 1,2