Properties

Label 68.3
Modulus $68$
Conductor $68$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(68, base_ring=CyclotomicField(16))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([8,1]))
 
pari: [g,chi] = znchar(Mod(3,68))
 

Basic properties

Modulus: \(68\)
Conductor: \(68\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
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 68.i

\(\chi_{68}(3,\cdot)\) \(\chi_{68}(7,\cdot)\) \(\chi_{68}(11,\cdot)\) \(\chi_{68}(23,\cdot)\) \(\chi_{68}(27,\cdot)\) \(\chi_{68}(31,\cdot)\) \(\chi_{68}(39,\cdot)\) \(\chi_{68}(63,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: \(\Q(\zeta_{68})^+\)

Values on generators

\((35,37)\) → \((-1,e\left(\frac{1}{16}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 68 }(3, a) \) \(1\)\(1\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{15}{16}\right)\)\(i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(e\left(\frac{7}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 68 }(3,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 68 }(3,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 68 }(3,·),\chi_{ 68 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 68 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2