Properties

Label 1024.9
Modulus $1024$
Conductor $512$
Order $128$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1024, base_ring=CyclotomicField(128))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,35]))
 
pari: [g,chi] = znchar(Mod(9,1024))
 

Basic properties

Modulus: \(1024\)
Conductor: \(512\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(128\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{512}(509,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1024.o

\(\chi_{1024}(9,\cdot)\) \(\chi_{1024}(25,\cdot)\) \(\chi_{1024}(41,\cdot)\) \(\chi_{1024}(57,\cdot)\) \(\chi_{1024}(73,\cdot)\) \(\chi_{1024}(89,\cdot)\) \(\chi_{1024}(105,\cdot)\) \(\chi_{1024}(121,\cdot)\) \(\chi_{1024}(137,\cdot)\) \(\chi_{1024}(153,\cdot)\) \(\chi_{1024}(169,\cdot)\) \(\chi_{1024}(185,\cdot)\) \(\chi_{1024}(201,\cdot)\) \(\chi_{1024}(217,\cdot)\) \(\chi_{1024}(233,\cdot)\) \(\chi_{1024}(249,\cdot)\) \(\chi_{1024}(265,\cdot)\) \(\chi_{1024}(281,\cdot)\) \(\chi_{1024}(297,\cdot)\) \(\chi_{1024}(313,\cdot)\) \(\chi_{1024}(329,\cdot)\) \(\chi_{1024}(345,\cdot)\) \(\chi_{1024}(361,\cdot)\) \(\chi_{1024}(377,\cdot)\) \(\chi_{1024}(393,\cdot)\) \(\chi_{1024}(409,\cdot)\) \(\chi_{1024}(425,\cdot)\) \(\chi_{1024}(441,\cdot)\) \(\chi_{1024}(457,\cdot)\) \(\chi_{1024}(473,\cdot)\) ...

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: $\Q(\zeta_{128})$
Fixed field: Number field defined by a degree 128 polynomial (not computed)

Values on generators

\((1023,5)\) → \((1,e\left(\frac{35}{128}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1024 }(9, a) \) \(1\)\(1\)\(e\left(\frac{73}{128}\right)\)\(e\left(\frac{35}{128}\right)\)\(e\left(\frac{15}{64}\right)\)\(e\left(\frac{9}{64}\right)\)\(e\left(\frac{31}{128}\right)\)\(e\left(\frac{45}{128}\right)\)\(e\left(\frac{27}{32}\right)\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{37}{128}\right)\)\(e\left(\frac{103}{128}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1024 }(9,a) \;\) at \(\;a = \) e.g. 2