Properties

Label 431.110
Modulus $431$
Conductor $431$
Order $43$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(431, base_ring=CyclotomicField(86))
 
M = H._module
 
chi = DirichletCharacter(H, M([24]))
 
pari: [g,chi] = znchar(Mod(110,431))
 

Basic properties

Modulus: \(431\)
Conductor: \(431\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(43\)
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 431.e

\(\chi_{431}(2,\cdot)\) \(\chi_{431}(3,\cdot)\) \(\chi_{431}(4,\cdot)\) \(\chi_{431}(6,\cdot)\) \(\chi_{431}(8,\cdot)\) \(\chi_{431}(9,\cdot)\) \(\chi_{431}(12,\cdot)\) \(\chi_{431}(16,\cdot)\) \(\chi_{431}(18,\cdot)\) \(\chi_{431}(24,\cdot)\) \(\chi_{431}(27,\cdot)\) \(\chi_{431}(32,\cdot)\) \(\chi_{431}(36,\cdot)\) \(\chi_{431}(48,\cdot)\) \(\chi_{431}(54,\cdot)\) \(\chi_{431}(55,\cdot)\) \(\chi_{431}(64,\cdot)\) \(\chi_{431}(72,\cdot)\) \(\chi_{431}(81,\cdot)\) \(\chi_{431}(96,\cdot)\) \(\chi_{431}(108,\cdot)\) \(\chi_{431}(110,\cdot)\) \(\chi_{431}(128,\cdot)\) \(\chi_{431}(144,\cdot)\) \(\chi_{431}(145,\cdot)\) \(\chi_{431}(149,\cdot)\) \(\chi_{431}(162,\cdot)\) \(\chi_{431}(165,\cdot)\) \(\chi_{431}(192,\cdot)\) \(\chi_{431}(216,\cdot)\) ...

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: $\Q(\zeta_{43})$
Fixed field: Number field defined by a degree 43 polynomial

Values on generators

\(7\) → \(e\left(\frac{12}{43}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 431 }(110, a) \) \(1\)\(1\)\(e\left(\frac{17}{43}\right)\)\(e\left(\frac{6}{43}\right)\)\(e\left(\frac{34}{43}\right)\)\(e\left(\frac{19}{43}\right)\)\(e\left(\frac{23}{43}\right)\)\(e\left(\frac{12}{43}\right)\)\(e\left(\frac{8}{43}\right)\)\(e\left(\frac{12}{43}\right)\)\(e\left(\frac{36}{43}\right)\)\(e\left(\frac{28}{43}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 431 }(110,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 431 }(110,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 431 }(110,·),\chi_{ 431 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 431 }(110,·)) \;\) at \(\; a,b = \) e.g. 1,2