Properties

Label 431.73
Modulus $431$
Conductor $431$
Order $430$
Real no
Primitive yes
Minimal yes
Parity odd

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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([361]))
 
pari: [g,chi] = znchar(Mod(73,431))
 

Basic properties

Modulus: \(431\)
Conductor: \(431\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(430\)
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 431.h

\(\chi_{431}(7,\cdot)\) \(\chi_{431}(13,\cdot)\) \(\chi_{431}(14,\cdot)\) \(\chi_{431}(17,\cdot)\) \(\chi_{431}(21,\cdot)\) \(\chi_{431}(28,\cdot)\) \(\chi_{431}(31,\cdot)\) \(\chi_{431}(34,\cdot)\) \(\chi_{431}(35,\cdot)\) \(\chi_{431}(37,\cdot)\) \(\chi_{431}(39,\cdot)\) \(\chi_{431}(42,\cdot)\) \(\chi_{431}(43,\cdot)\) \(\chi_{431}(51,\cdot)\) \(\chi_{431}(52,\cdot)\) \(\chi_{431}(56,\cdot)\) \(\chi_{431}(62,\cdot)\) \(\chi_{431}(63,\cdot)\) \(\chi_{431}(65,\cdot)\) \(\chi_{431}(67,\cdot)\) \(\chi_{431}(68,\cdot)\) \(\chi_{431}(70,\cdot)\) \(\chi_{431}(71,\cdot)\) \(\chi_{431}(73,\cdot)\) \(\chi_{431}(74,\cdot)\) \(\chi_{431}(77,\cdot)\) \(\chi_{431}(78,\cdot)\) \(\chi_{431}(79,\cdot)\) \(\chi_{431}(83,\cdot)\) \(\chi_{431}(84,\cdot)\) ...

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: $\Q(\zeta_{215})$
Fixed field: Number field defined by a degree 430 polynomial (not computed)

Values on generators

\(7\) → \(e\left(\frac{361}{430}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 431 }(73, a) \) \(-1\)\(1\)\(e\left(\frac{30}{43}\right)\)\(e\left(\frac{3}{43}\right)\)\(e\left(\frac{17}{43}\right)\)\(e\left(\frac{26}{215}\right)\)\(e\left(\frac{33}{43}\right)\)\(e\left(\frac{361}{430}\right)\)\(e\left(\frac{4}{43}\right)\)\(e\left(\frac{6}{43}\right)\)\(e\left(\frac{176}{215}\right)\)\(e\left(\frac{199}{215}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 431 }(73,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 431 }(73,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 431 }(73,·),\chi_{ 431 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 431 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2