Properties

Label 162.7
Modulus $162$
Conductor $81$
Order $27$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(162, base_ring=CyclotomicField(54))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([16]))
 
pari: [g,chi] = znchar(Mod(7,162))
 

Basic properties

Modulus: \(162\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(27\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 162.g

\(\chi_{162}(7,\cdot)\) \(\chi_{162}(13,\cdot)\) \(\chi_{162}(25,\cdot)\) \(\chi_{162}(31,\cdot)\) \(\chi_{162}(43,\cdot)\) \(\chi_{162}(49,\cdot)\) \(\chi_{162}(61,\cdot)\) \(\chi_{162}(67,\cdot)\) \(\chi_{162}(79,\cdot)\) \(\chi_{162}(85,\cdot)\) \(\chi_{162}(97,\cdot)\) \(\chi_{162}(103,\cdot)\) \(\chi_{162}(115,\cdot)\) \(\chi_{162}(121,\cdot)\) \(\chi_{162}(133,\cdot)\) \(\chi_{162}(139,\cdot)\) \(\chi_{162}(151,\cdot)\) \(\chi_{162}(157,\cdot)\)

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

\(83\) → \(e\left(\frac{8}{27}\right)\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 162 }(7, a) \) \(1\)\(1\)\(e\left(\frac{22}{27}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{17}{27}\right)\)\(e\left(\frac{26}{27}\right)\)\(e\left(\frac{25}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 162 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 162 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 162 }(7,·),\chi_{ 162 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 162 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2