Properties

Label 163.71
Modulus $163$
Conductor $163$
Order $81$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(163, base_ring=CyclotomicField(162))
 
M = H._module
 
chi = DirichletCharacter(H, M([92]))
 
pari: [g,chi] = znchar(Mod(71,163))
 

Basic properties

Modulus: \(163\)
Conductor: \(163\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(81\)
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 163.i

\(\chi_{163}(4,\cdot)\) \(\chi_{163}(9,\cdot)\) \(\chi_{163}(10,\cdot)\) \(\chi_{163}(14,\cdot)\) \(\chi_{163}(15,\cdot)\) \(\chi_{163}(16,\cdot)\) \(\chi_{163}(24,\cdot)\) \(\chi_{163}(26,\cdot)\) \(\chi_{163}(33,\cdot)\) \(\chi_{163}(34,\cdot)\) \(\chi_{163}(35,\cdot)\) \(\chi_{163}(39,\cdot)\) \(\chi_{163}(41,\cdot)\) \(\chi_{163}(43,\cdot)\) \(\chi_{163}(46,\cdot)\) \(\chi_{163}(47,\cdot)\) \(\chi_{163}(49,\cdot)\) \(\chi_{163}(51,\cdot)\) \(\chi_{163}(54,\cdot)\) \(\chi_{163}(55,\cdot)\) \(\chi_{163}(56,\cdot)\) \(\chi_{163}(57,\cdot)\) \(\chi_{163}(60,\cdot)\) \(\chi_{163}(62,\cdot)\) \(\chi_{163}(69,\cdot)\) \(\chi_{163}(71,\cdot)\) \(\chi_{163}(74,\cdot)\) \(\chi_{163}(81,\cdot)\) \(\chi_{163}(83,\cdot)\) \(\chi_{163}(84,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{81})$
Fixed field: Number field defined by a degree 81 polynomial

Values on generators

\(2\) → \(e\left(\frac{46}{81}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 163 }(71, a) \) \(1\)\(1\)\(e\left(\frac{46}{81}\right)\)\(e\left(\frac{29}{81}\right)\)\(e\left(\frac{11}{81}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{37}{81}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{58}{81}\right)\)\(e\left(\frac{7}{81}\right)\)\(e\left(\frac{56}{81}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 163 }(71,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 163 }(71,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 163 }(71,·),\chi_{ 163 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 163 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2