Properties

Label 131.10
Modulus $131$
Conductor $131$
Order $130$
Real no
Primitive yes
Minimal yes
Parity odd

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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([47]))
 
pari: [g,chi] = znchar(Mod(10,131))
 

Basic properties

Modulus: \(131\)
Conductor: \(131\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(130\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 131.h

\(\chi_{131}(2,\cdot)\) \(\chi_{131}(6,\cdot)\) \(\chi_{131}(8,\cdot)\) \(\chi_{131}(10,\cdot)\) \(\chi_{131}(14,\cdot)\) \(\chi_{131}(17,\cdot)\) \(\chi_{131}(22,\cdot)\) \(\chi_{131}(23,\cdot)\) \(\chi_{131}(26,\cdot)\) \(\chi_{131}(29,\cdot)\) \(\chi_{131}(30,\cdot)\) \(\chi_{131}(31,\cdot)\) \(\chi_{131}(37,\cdot)\) \(\chi_{131}(40,\cdot)\) \(\chi_{131}(50,\cdot)\) \(\chi_{131}(54,\cdot)\) \(\chi_{131}(56,\cdot)\) \(\chi_{131}(57,\cdot)\) \(\chi_{131}(66,\cdot)\) \(\chi_{131}(67,\cdot)\) \(\chi_{131}(72,\cdot)\) \(\chi_{131}(76,\cdot)\) \(\chi_{131}(82,\cdot)\) \(\chi_{131}(83,\cdot)\) \(\chi_{131}(85,\cdot)\) \(\chi_{131}(87,\cdot)\) \(\chi_{131}(88,\cdot)\) \(\chi_{131}(90,\cdot)\) \(\chi_{131}(93,\cdot)\) \(\chi_{131}(95,\cdot)\) ...

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 130 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{47}{130}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 131 }(10, a) \) \(-1\)\(1\)\(e\left(\frac{47}{130}\right)\)\(e\left(\frac{2}{65}\right)\)\(e\left(\frac{47}{65}\right)\)\(e\left(\frac{41}{65}\right)\)\(e\left(\frac{51}{130}\right)\)\(e\left(\frac{46}{65}\right)\)\(e\left(\frac{11}{130}\right)\)\(e\left(\frac{4}{65}\right)\)\(e\left(\frac{129}{130}\right)\)\(e\left(\frac{16}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 131 }(10,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 131 }(10,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 131 }(10,·),\chi_{ 131 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 131 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2