Properties

Label 714.25
Modulus $714$
Conductor $119$
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([0,16,15]))
 
pari: [g,chi] = znchar(Mod(25,714))
 

Basic properties

Modulus: \(714\)
Conductor: \(119\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{119}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 714.bh

\(\chi_{714}(25,\cdot)\) \(\chi_{714}(121,\cdot)\) \(\chi_{714}(151,\cdot)\) \(\chi_{714}(247,\cdot)\) \(\chi_{714}(331,\cdot)\) \(\chi_{714}(457,\cdot)\) \(\chi_{714}(529,\cdot)\) \(\chi_{714}(655,\cdot)\)

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: 24.24.2296127442650479958000916502307873630417.1

Values on generators

\((239,409,547)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 714 }(25, a) \) \(1\)\(1\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{1}{24}\right)\)\(-1\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 714 }(25,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 714 }(25,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 714 }(25,·),\chi_{ 714 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 714 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2