Properties

Label 162.113
Modulus $162$
Conductor $81$
Order $54$
Real no
Primitive no
Minimal yes
Parity odd

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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([5]))
 
pari: [g,chi] = znchar(Mod(113,162))
 

Basic properties

Modulus: \(162\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{81}(32,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 162.h

\(\chi_{162}(5,\cdot)\) \(\chi_{162}(11,\cdot)\) \(\chi_{162}(23,\cdot)\) \(\chi_{162}(29,\cdot)\) \(\chi_{162}(41,\cdot)\) \(\chi_{162}(47,\cdot)\) \(\chi_{162}(59,\cdot)\) \(\chi_{162}(65,\cdot)\) \(\chi_{162}(77,\cdot)\) \(\chi_{162}(83,\cdot)\) \(\chi_{162}(95,\cdot)\) \(\chi_{162}(101,\cdot)\) \(\chi_{162}(113,\cdot)\) \(\chi_{162}(119,\cdot)\) \(\chi_{162}(131,\cdot)\) \(\chi_{162}(137,\cdot)\) \(\chi_{162}(149,\cdot)\) \(\chi_{162}(155,\cdot)\)

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: Number field defined by a degree 54 polynomial

Values on generators

\(83\) → \(e\left(\frac{5}{54}\right)\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 162 }(113, a) \) \(-1\)\(1\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{11}{54}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{54}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{23}{27}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 162 }(113,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 162 }(113,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 162 }(113,·),\chi_{ 162 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 162 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2