Properties

Label 256.19
Modulus $256$
Conductor $256$
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(256, base_ring=CyclotomicField(64))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,23]))
 
pari: [g,chi] = znchar(Mod(19,256))
 

Basic properties

Modulus: \(256\)
Conductor: \(256\)
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 256.n

\(\chi_{256}(3,\cdot)\) \(\chi_{256}(11,\cdot)\) \(\chi_{256}(19,\cdot)\) \(\chi_{256}(27,\cdot)\) \(\chi_{256}(35,\cdot)\) \(\chi_{256}(43,\cdot)\) \(\chi_{256}(51,\cdot)\) \(\chi_{256}(59,\cdot)\) \(\chi_{256}(67,\cdot)\) \(\chi_{256}(75,\cdot)\) \(\chi_{256}(83,\cdot)\) \(\chi_{256}(91,\cdot)\) \(\chi_{256}(99,\cdot)\) \(\chi_{256}(107,\cdot)\) \(\chi_{256}(115,\cdot)\) \(\chi_{256}(123,\cdot)\) \(\chi_{256}(131,\cdot)\) \(\chi_{256}(139,\cdot)\) \(\chi_{256}(147,\cdot)\) \(\chi_{256}(155,\cdot)\) \(\chi_{256}(163,\cdot)\) \(\chi_{256}(171,\cdot)\) \(\chi_{256}(179,\cdot)\) \(\chi_{256}(187,\cdot)\) \(\chi_{256}(195,\cdot)\) \(\chi_{256}(203,\cdot)\) \(\chi_{256}(211,\cdot)\) \(\chi_{256}(219,\cdot)\) \(\chi_{256}(227,\cdot)\) \(\chi_{256}(235,\cdot)\) ...

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

Values on generators

\((255,5)\) → \((-1,e\left(\frac{23}{64}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 256 }(19, a) \) \(-1\)\(1\)\(e\left(\frac{5}{64}\right)\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{49}{64}\right)\)\(e\left(\frac{11}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 256 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 256 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 256 }(19,·),\chi_{ 256 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 256 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2