Properties

Label 256.37
Modulus $256$
Conductor $256$
Order $64$
Real no
Primitive yes
Minimal yes
Parity even

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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([0,25]))
 
pari: [g,chi] = znchar(Mod(37,256))
 

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

Galois orbit 256.m

\(\chi_{256}(5,\cdot)\) \(\chi_{256}(13,\cdot)\) \(\chi_{256}(21,\cdot)\) \(\chi_{256}(29,\cdot)\) \(\chi_{256}(37,\cdot)\) \(\chi_{256}(45,\cdot)\) \(\chi_{256}(53,\cdot)\) \(\chi_{256}(61,\cdot)\) \(\chi_{256}(69,\cdot)\) \(\chi_{256}(77,\cdot)\) \(\chi_{256}(85,\cdot)\) \(\chi_{256}(93,\cdot)\) \(\chi_{256}(101,\cdot)\) \(\chi_{256}(109,\cdot)\) \(\chi_{256}(117,\cdot)\) \(\chi_{256}(125,\cdot)\) \(\chi_{256}(133,\cdot)\) \(\chi_{256}(141,\cdot)\) \(\chi_{256}(149,\cdot)\) \(\chi_{256}(157,\cdot)\) \(\chi_{256}(165,\cdot)\) \(\chi_{256}(173,\cdot)\) \(\chi_{256}(181,\cdot)\) \(\chi_{256}(189,\cdot)\) \(\chi_{256}(197,\cdot)\) \(\chi_{256}(205,\cdot)\) \(\chi_{256}(213,\cdot)\) \(\chi_{256}(221,\cdot)\) \(\chi_{256}(229,\cdot)\) \(\chi_{256}(237,\cdot)\) ...

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

\((255,5)\) → \((1,e\left(\frac{25}{64}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 256 }(37, a) \) \(1\)\(1\)\(e\left(\frac{43}{64}\right)\)\(e\left(\frac{25}{64}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{11}{32}\right)\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{63}{64}\right)\)\(e\left(\frac{37}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 256 }(37,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 256 }(37,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 256 }(37,·),\chi_{ 256 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 256 }(37,·)) \;\) at \(\; a,b = \) e.g. 1,2