Properties

Label 431.41
Modulus $431$
Conductor $431$
Order $215$
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(430))
 
M = H._module
 
chi = DirichletCharacter(H, M([364]))
 
pari: [g,chi] = znchar(Mod(41,431))
 

Basic properties

Modulus: \(431\)
Conductor: \(431\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(215\)
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.g

\(\chi_{431}(5,\cdot)\) \(\chi_{431}(10,\cdot)\) \(\chi_{431}(11,\cdot)\) \(\chi_{431}(15,\cdot)\) \(\chi_{431}(19,\cdot)\) \(\chi_{431}(20,\cdot)\) \(\chi_{431}(22,\cdot)\) \(\chi_{431}(23,\cdot)\) \(\chi_{431}(25,\cdot)\) \(\chi_{431}(29,\cdot)\) \(\chi_{431}(30,\cdot)\) \(\chi_{431}(33,\cdot)\) \(\chi_{431}(38,\cdot)\) \(\chi_{431}(40,\cdot)\) \(\chi_{431}(41,\cdot)\) \(\chi_{431}(44,\cdot)\) \(\chi_{431}(45,\cdot)\) \(\chi_{431}(46,\cdot)\) \(\chi_{431}(49,\cdot)\) \(\chi_{431}(50,\cdot)\) \(\chi_{431}(53,\cdot)\) \(\chi_{431}(57,\cdot)\) \(\chi_{431}(58,\cdot)\) \(\chi_{431}(59,\cdot)\) \(\chi_{431}(60,\cdot)\) \(\chi_{431}(61,\cdot)\) \(\chi_{431}(66,\cdot)\) \(\chi_{431}(69,\cdot)\) \(\chi_{431}(75,\cdot)\) \(\chi_{431}(76,\cdot)\) ...

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: $\Q(\zeta_{215})$
Fixed field: Number field defined by a degree 215 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{182}{215}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 431 }(41, a) \) \(1\)\(1\)\(e\left(\frac{10}{43}\right)\)\(e\left(\frac{1}{43}\right)\)\(e\left(\frac{20}{43}\right)\)\(e\left(\frac{109}{215}\right)\)\(e\left(\frac{11}{43}\right)\)\(e\left(\frac{182}{215}\right)\)\(e\left(\frac{30}{43}\right)\)\(e\left(\frac{2}{43}\right)\)\(e\left(\frac{159}{215}\right)\)\(e\left(\frac{181}{215}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 431 }(41,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 431 }(41,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 431 }(41,·),\chi_{ 431 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 431 }(41,·)) \;\) at \(\; a,b = \) e.g. 1,2