Properties

Label 512.11
Modulus $512$
Conductor $512$
Order $128$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(512, base_ring=CyclotomicField(128))
 
M = H._module
 
chi = DirichletCharacter(H, M([64,85]))
 
pari: [g,chi] = znchar(Mod(11,512))
 

Basic properties

Modulus: \(512\)
Conductor: \(512\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(128\)
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 512.p

\(\chi_{512}(3,\cdot)\) \(\chi_{512}(11,\cdot)\) \(\chi_{512}(19,\cdot)\) \(\chi_{512}(27,\cdot)\) \(\chi_{512}(35,\cdot)\) \(\chi_{512}(43,\cdot)\) \(\chi_{512}(51,\cdot)\) \(\chi_{512}(59,\cdot)\) \(\chi_{512}(67,\cdot)\) \(\chi_{512}(75,\cdot)\) \(\chi_{512}(83,\cdot)\) \(\chi_{512}(91,\cdot)\) \(\chi_{512}(99,\cdot)\) \(\chi_{512}(107,\cdot)\) \(\chi_{512}(115,\cdot)\) \(\chi_{512}(123,\cdot)\) \(\chi_{512}(131,\cdot)\) \(\chi_{512}(139,\cdot)\) \(\chi_{512}(147,\cdot)\) \(\chi_{512}(155,\cdot)\) \(\chi_{512}(163,\cdot)\) \(\chi_{512}(171,\cdot)\) \(\chi_{512}(179,\cdot)\) \(\chi_{512}(187,\cdot)\) \(\chi_{512}(195,\cdot)\) \(\chi_{512}(203,\cdot)\) \(\chi_{512}(211,\cdot)\) \(\chi_{512}(219,\cdot)\) \(\chi_{512}(227,\cdot)\) \(\chi_{512}(235,\cdot)\) ...

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: $\Q(\zeta_{128})$
Fixed field: Number field defined by a degree 128 polynomial (not computed)

Values on generators

\((511,5)\) → \((-1,e\left(\frac{85}{128}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 512 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{95}{128}\right)\)\(e\left(\frac{85}{128}\right)\)\(e\left(\frac{41}{64}\right)\)\(e\left(\frac{31}{64}\right)\)\(e\left(\frac{121}{128}\right)\)\(e\left(\frac{91}{128}\right)\)\(e\left(\frac{13}{32}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{99}{128}\right)\)\(e\left(\frac{49}{128}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 512 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 512 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 512 }(11,·),\chi_{ 512 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 512 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2