Properties

Label 128.69
Modulus $128$
Conductor $128$
Order $32$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,17]))
 
pari: [g,chi] = znchar(Mod(69,128))
 

Basic properties

Modulus: \(128\)
Conductor: \(128\)
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 128.k

\(\chi_{128}(5,\cdot)\) \(\chi_{128}(13,\cdot)\) \(\chi_{128}(21,\cdot)\) \(\chi_{128}(29,\cdot)\) \(\chi_{128}(37,\cdot)\) \(\chi_{128}(45,\cdot)\) \(\chi_{128}(53,\cdot)\) \(\chi_{128}(61,\cdot)\) \(\chi_{128}(69,\cdot)\) \(\chi_{128}(77,\cdot)\) \(\chi_{128}(85,\cdot)\) \(\chi_{128}(93,\cdot)\) \(\chi_{128}(101,\cdot)\) \(\chi_{128}(109,\cdot)\) \(\chi_{128}(117,\cdot)\) \(\chi_{128}(125,\cdot)\)

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

\((127,5)\) → \((1,e\left(\frac{17}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 128 }(69, a) \) \(1\)\(1\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{31}{32}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{29}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 128 }(69,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 128 }(69,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 128 }(69,·),\chi_{ 128 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 128 }(69,·)) \;\) at \(\; a,b = \) e.g. 1,2