Properties

Label 128.3
Modulus $128$
Conductor $128$
Order $32$
Real no
Primitive yes
Minimal yes
Parity odd

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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([16,3]))
 
pari: [g,chi] = znchar(Mod(3,128))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 128.l

\(\chi_{128}(3,\cdot)\) \(\chi_{128}(11,\cdot)\) \(\chi_{128}(19,\cdot)\) \(\chi_{128}(27,\cdot)\) \(\chi_{128}(35,\cdot)\) \(\chi_{128}(43,\cdot)\) \(\chi_{128}(51,\cdot)\) \(\chi_{128}(59,\cdot)\) \(\chi_{128}(67,\cdot)\) \(\chi_{128}(75,\cdot)\) \(\chi_{128}(83,\cdot)\) \(\chi_{128}(91,\cdot)\) \(\chi_{128}(99,\cdot)\) \(\chi_{128}(107,\cdot)\) \(\chi_{128}(115,\cdot)\) \(\chi_{128}(123,\cdot)\)

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: 32.0.3138550867693340381917894711603833208051177722232017256448.1

Values on generators

\((127,5)\) → \((-1,e\left(\frac{3}{32}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 128 }(3, a) \) \(-1\)\(1\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{7}{32}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 128 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 128 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 128 }(3,·),\chi_{ 128 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 128 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2