Properties

Label 768.269
Modulus $768$
Conductor $768$
Order $64$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,47,32]))
 
pari: [g,chi] = znchar(Mod(269,768))
 

Basic properties

Modulus: \(768\)
Conductor: \(768\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
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 768.y

\(\chi_{768}(5,\cdot)\) \(\chi_{768}(29,\cdot)\) \(\chi_{768}(53,\cdot)\) \(\chi_{768}(77,\cdot)\) \(\chi_{768}(101,\cdot)\) \(\chi_{768}(125,\cdot)\) \(\chi_{768}(149,\cdot)\) \(\chi_{768}(173,\cdot)\) \(\chi_{768}(197,\cdot)\) \(\chi_{768}(221,\cdot)\) \(\chi_{768}(245,\cdot)\) \(\chi_{768}(269,\cdot)\) \(\chi_{768}(293,\cdot)\) \(\chi_{768}(317,\cdot)\) \(\chi_{768}(341,\cdot)\) \(\chi_{768}(365,\cdot)\) \(\chi_{768}(389,\cdot)\) \(\chi_{768}(413,\cdot)\) \(\chi_{768}(437,\cdot)\) \(\chi_{768}(461,\cdot)\) \(\chi_{768}(485,\cdot)\) \(\chi_{768}(509,\cdot)\) \(\chi_{768}(533,\cdot)\) \(\chi_{768}(557,\cdot)\) \(\chi_{768}(581,\cdot)\) \(\chi_{768}(605,\cdot)\) \(\chi_{768}(629,\cdot)\) \(\chi_{768}(653,\cdot)\) \(\chi_{768}(677,\cdot)\) \(\chi_{768}(701,\cdot)\) ...

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: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial

Values on generators

\((511,517,257)\) → \((1,e\left(\frac{47}{64}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 768 }(269, a) \) \(-1\)\(1\)\(e\left(\frac{15}{64}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{59}{64}\right)\)\(e\left(\frac{33}{64}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{53}{64}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 768 }(269,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 768 }(269,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 768 }(269,·),\chi_{ 768 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 768 }(269,·)) \;\) at \(\; a,b = \) e.g. 1,2