Properties

Label 256.5
Modulus $256$
Conductor $256$
Order $64$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(256)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(5,256))
 

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()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{35}{64}\right)\)\(e\left(\frac{1}{64}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{3}{32}\right)\)\(e\left(\frac{21}{64}\right)\)\(e\left(\frac{47}{64}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{45}{64}\right)\)
value at e.g. 2

Related number fields

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

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 256 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{256}(5,\cdot)) = \sum_{r\in \Z/256\Z} \chi_{256}(5,r) e\left(\frac{r}{128}\right) = -0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 256 }(5,·),\chi_{ 256 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{256}(5,\cdot),\chi_{256}(1,\cdot)) = \sum_{r\in \Z/256\Z} \chi_{256}(5,r) \chi_{256}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 256 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{256}(5,·)) = \sum_{r \in \Z/256\Z} \chi_{256}(5,r) e\left(\frac{1 r + 2 r^{-1}}{256}\right) = -12.1153415441+-10.4507654873i \)