Properties

Label 81.70
Modulus $81$
Conductor $81$
Order $27$
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(81, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([40]))
 
pari: [g,chi] = znchar(Mod(70,81))
 

Basic properties

Modulus: \(81\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(27\)
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 81.g

\(\chi_{81}(4,\cdot)\) \(\chi_{81}(7,\cdot)\) \(\chi_{81}(13,\cdot)\) \(\chi_{81}(16,\cdot)\) \(\chi_{81}(22,\cdot)\) \(\chi_{81}(25,\cdot)\) \(\chi_{81}(31,\cdot)\) \(\chi_{81}(34,\cdot)\) \(\chi_{81}(40,\cdot)\) \(\chi_{81}(43,\cdot)\) \(\chi_{81}(49,\cdot)\) \(\chi_{81}(52,\cdot)\) \(\chi_{81}(58,\cdot)\) \(\chi_{81}(61,\cdot)\) \(\chi_{81}(67,\cdot)\) \(\chi_{81}(70,\cdot)\) \(\chi_{81}(76,\cdot)\) \(\chi_{81}(79,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: \(\Q(\zeta_{81})^+\)

Values on generators

\(2\) → \(e\left(\frac{20}{27}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 81 }(70, a) \) \(1\)\(1\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{1}{27}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{26}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 81 }(70,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 81 }(70,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 81 }(70,·),\chi_{ 81 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 81 }(70,·)) \;\) at \(\; a,b = \) e.g. 1,2