Properties

Label 256.17
Modulus $256$
Conductor $64$
Order $16$
Real no
Primitive no
Minimal no
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([0,7]))
 
pari: [g,chi] = znchar(Mod(17,256))
 

Basic properties

Modulus: \(256\)
Conductor: \(64\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{64}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 256.i

\(\chi_{256}(17,\cdot)\) \(\chi_{256}(49,\cdot)\) \(\chi_{256}(81,\cdot)\) \(\chi_{256}(113,\cdot)\) \(\chi_{256}(145,\cdot)\) \(\chi_{256}(177,\cdot)\) \(\chi_{256}(209,\cdot)\) \(\chi_{256}(241,\cdot)\)

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

\((255,5)\) → \((1,e\left(\frac{7}{16}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{9}{16}\right)\)\(-i\)\(i\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{11}{16}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{64})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 256 }(17,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{256}(17,\cdot)) = \sum_{r\in \Z/256\Z} \chi_{256}(17,r) e\left(\frac{r}{128}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 256 }(17,·),\chi_{ 256 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{256}(17,\cdot),\chi_{256}(1,\cdot)) = \sum_{r\in \Z/256\Z} \chi_{256}(17,r) \chi_{256}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 256 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{256}(17,·)) = \sum_{r \in \Z/256\Z} \chi_{256}(17,r) e\left(\frac{1 r + 2 r^{-1}}{256}\right) = -0.0 \)