Properties

Label 131.46
Modulus $131$
Conductor $131$
Order $65$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(131, base_ring=CyclotomicField(130))
 
M = H._module
 
chi = DirichletCharacter(H, M([24]))
 
pari: [g,chi] = znchar(Mod(46,131))
 

Basic properties

Modulus: \(131\)
Conductor: \(131\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(65\)
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 131.g

\(\chi_{131}(3,\cdot)\) \(\chi_{131}(4,\cdot)\) \(\chi_{131}(5,\cdot)\) \(\chi_{131}(7,\cdot)\) \(\chi_{131}(9,\cdot)\) \(\chi_{131}(11,\cdot)\) \(\chi_{131}(12,\cdot)\) \(\chi_{131}(13,\cdot)\) \(\chi_{131}(15,\cdot)\) \(\chi_{131}(16,\cdot)\) \(\chi_{131}(20,\cdot)\) \(\chi_{131}(21,\cdot)\) \(\chi_{131}(25,\cdot)\) \(\chi_{131}(27,\cdot)\) \(\chi_{131}(28,\cdot)\) \(\chi_{131}(33,\cdot)\) \(\chi_{131}(34,\cdot)\) \(\chi_{131}(35,\cdot)\) \(\chi_{131}(36,\cdot)\) \(\chi_{131}(38,\cdot)\) \(\chi_{131}(41,\cdot)\) \(\chi_{131}(43,\cdot)\) \(\chi_{131}(44,\cdot)\) \(\chi_{131}(46,\cdot)\) \(\chi_{131}(48,\cdot)\) \(\chi_{131}(49,\cdot)\) \(\chi_{131}(55,\cdot)\) \(\chi_{131}(59,\cdot)\) \(\chi_{131}(64,\cdot)\) \(\chi_{131}(65,\cdot)\) ...

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: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 65 polynomial

Values on generators

\(2\) → \(e\left(\frac{12}{65}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 131 }(46, a) \) \(1\)\(1\)\(e\left(\frac{12}{65}\right)\)\(e\left(\frac{19}{65}\right)\)\(e\left(\frac{24}{65}\right)\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{47}{65}\right)\)\(e\left(\frac{36}{65}\right)\)\(e\left(\frac{38}{65}\right)\)\(e\left(\frac{44}{65}\right)\)\(e\left(\frac{22}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 131 }(46,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 131 }(46,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 131 }(46,·),\chi_{ 131 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 131 }(46,·)) \;\) at \(\; a,b = \) e.g. 1,2