Properties

Label 862.113
Modulus $862$
Conductor $431$
Order $430$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(862, base_ring=CyclotomicField(430))
 
M = H._module
 
chi = DirichletCharacter(H, M([271]))
 
pari: [g,chi] = znchar(Mod(113,862))
 

Basic properties

Modulus: \(862\)
Conductor: \(431\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(430\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{431}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 862.h

\(\chi_{862}(7,\cdot)\) \(\chi_{862}(13,\cdot)\) \(\chi_{862}(17,\cdot)\) \(\chi_{862}(21,\cdot)\) \(\chi_{862}(31,\cdot)\) \(\chi_{862}(35,\cdot)\) \(\chi_{862}(37,\cdot)\) \(\chi_{862}(39,\cdot)\) \(\chi_{862}(43,\cdot)\) \(\chi_{862}(51,\cdot)\) \(\chi_{862}(63,\cdot)\) \(\chi_{862}(65,\cdot)\) \(\chi_{862}(67,\cdot)\) \(\chi_{862}(71,\cdot)\) \(\chi_{862}(73,\cdot)\) \(\chi_{862}(77,\cdot)\) \(\chi_{862}(79,\cdot)\) \(\chi_{862}(83,\cdot)\) \(\chi_{862}(85,\cdot)\) \(\chi_{862}(89,\cdot)\) \(\chi_{862}(93,\cdot)\) \(\chi_{862}(103,\cdot)\) \(\chi_{862}(105,\cdot)\) \(\chi_{862}(111,\cdot)\) \(\chi_{862}(113,\cdot)\) \(\chi_{862}(117,\cdot)\) \(\chi_{862}(127,\cdot)\) \(\chi_{862}(129,\cdot)\) \(\chi_{862}(131,\cdot)\) \(\chi_{862}(137,\cdot)\) ...

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

\(7\) → \(e\left(\frac{271}{430}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 862 }(113, a) \) \(-1\)\(1\)\(e\left(\frac{20}{43}\right)\)\(e\left(\frac{116}{215}\right)\)\(e\left(\frac{271}{430}\right)\)\(e\left(\frac{40}{43}\right)\)\(e\left(\frac{94}{215}\right)\)\(e\left(\frac{107}{430}\right)\)\(e\left(\frac{1}{215}\right)\)\(e\left(\frac{231}{430}\right)\)\(e\left(\frac{142}{215}\right)\)\(e\left(\frac{41}{430}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 862 }(113,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 862 }(113,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 862 }(113,·),\chi_{ 862 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 862 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2