Properties

Label 768.71
Modulus $768$
Conductor $384$
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(768, base_ring=CyclotomicField(32))
 
M = H._module
 
chi = DirichletCharacter(H, M([16,13,16]))
 
pari: [g,chi] = znchar(Mod(71,768))
 

Basic properties

Modulus: \(768\)
Conductor: \(384\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(32\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{384}(107,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 768.w

\(\chi_{768}(23,\cdot)\) \(\chi_{768}(71,\cdot)\) \(\chi_{768}(119,\cdot)\) \(\chi_{768}(167,\cdot)\) \(\chi_{768}(215,\cdot)\) \(\chi_{768}(263,\cdot)\) \(\chi_{768}(311,\cdot)\) \(\chi_{768}(359,\cdot)\) \(\chi_{768}(407,\cdot)\) \(\chi_{768}(455,\cdot)\) \(\chi_{768}(503,\cdot)\) \(\chi_{768}(551,\cdot)\) \(\chi_{768}(599,\cdot)\) \(\chi_{768}(647,\cdot)\) \(\chi_{768}(695,\cdot)\) \(\chi_{768}(743,\cdot)\)

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: 32.32.135104323545903136978453058557785670637514001130337144105502507008.1

Values on generators

\((511,517,257)\) → \((-1,e\left(\frac{13}{32}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 768 }(71, a) \) \(1\)\(1\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{17}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{15}{32}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 768 }(71,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 768 }(71,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 768 }(71,·),\chi_{ 768 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 768 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2