Properties

Label 512.9
Modulus $512$
Conductor $256$
Order $64$
Real no
Primitive no
Minimal no
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(512, base_ring=CyclotomicField(64))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,35]))
 
pari: [g,chi] = znchar(Mod(9,512))
 

Basic properties

Modulus: \(512\)
Conductor: \(256\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{256}(253,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 512.m

\(\chi_{512}(9,\cdot)\) \(\chi_{512}(25,\cdot)\) \(\chi_{512}(41,\cdot)\) \(\chi_{512}(57,\cdot)\) \(\chi_{512}(73,\cdot)\) \(\chi_{512}(89,\cdot)\) \(\chi_{512}(105,\cdot)\) \(\chi_{512}(121,\cdot)\) \(\chi_{512}(137,\cdot)\) \(\chi_{512}(153,\cdot)\) \(\chi_{512}(169,\cdot)\) \(\chi_{512}(185,\cdot)\) \(\chi_{512}(201,\cdot)\) \(\chi_{512}(217,\cdot)\) \(\chi_{512}(233,\cdot)\) \(\chi_{512}(249,\cdot)\) \(\chi_{512}(265,\cdot)\) \(\chi_{512}(281,\cdot)\) \(\chi_{512}(297,\cdot)\) \(\chi_{512}(313,\cdot)\) \(\chi_{512}(329,\cdot)\) \(\chi_{512}(345,\cdot)\) \(\chi_{512}(361,\cdot)\) \(\chi_{512}(377,\cdot)\) \(\chi_{512}(393,\cdot)\) \(\chi_{512}(409,\cdot)\) \(\chi_{512}(425,\cdot)\) \(\chi_{512}(441,\cdot)\) \(\chi_{512}(457,\cdot)\) \(\chi_{512}(473,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((511,5)\) → \((1,e\left(\frac{35}{64}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{9}{64}\right)\)\(e\left(\frac{35}{64}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{31}{64}\right)\)\(e\left(\frac{45}{64}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{37}{64}\right)\)\(e\left(\frac{39}{64}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 512 }(9,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 512 }(9,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 512 }(9,·),\chi_{ 512 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 512 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2