Properties

Label 714.59
Modulus $714$
Conductor $357$
Order $24$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,4,15]))
 
pari: [g,chi] = znchar(Mod(59,714))
 

Basic properties

Modulus: \(714\)
Conductor: \(357\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{357}(59,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 714.bj

\(\chi_{714}(59,\cdot)\) \(\chi_{714}(185,\cdot)\) \(\chi_{714}(257,\cdot)\) \(\chi_{714}(383,\cdot)\) \(\chi_{714}(467,\cdot)\) \(\chi_{714}(563,\cdot)\) \(\chi_{714}(593,\cdot)\) \(\chi_{714}(689,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: Number field defined by a degree 24 polynomial

Values on generators

\((239,409,547)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 714 }(59, a) \) \(1\)\(1\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{13}{24}\right)\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 714 }(59,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 714 }(59,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 714 }(59,·),\chi_{ 714 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 714 }(59,·)) \;\) at \(\; a,b = \) e.g. 1,2