Properties

Label 784.299
Modulus $784$
Conductor $784$
Order $84$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,21,58]))
 
pari: [g,chi] = znchar(Mod(299,784))
 

Basic properties

Modulus: \(784\)
Conductor: \(784\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
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 784.bu

\(\chi_{784}(3,\cdot)\) \(\chi_{784}(59,\cdot)\) \(\chi_{784}(75,\cdot)\) \(\chi_{784}(115,\cdot)\) \(\chi_{784}(131,\cdot)\) \(\chi_{784}(171,\cdot)\) \(\chi_{784}(187,\cdot)\) \(\chi_{784}(243,\cdot)\) \(\chi_{784}(283,\cdot)\) \(\chi_{784}(299,\cdot)\) \(\chi_{784}(339,\cdot)\) \(\chi_{784}(355,\cdot)\) \(\chi_{784}(395,\cdot)\) \(\chi_{784}(451,\cdot)\) \(\chi_{784}(467,\cdot)\) \(\chi_{784}(507,\cdot)\) \(\chi_{784}(523,\cdot)\) \(\chi_{784}(563,\cdot)\) \(\chi_{784}(579,\cdot)\) \(\chi_{784}(635,\cdot)\) \(\chi_{784}(675,\cdot)\) \(\chi_{784}(691,\cdot)\) \(\chi_{784}(731,\cdot)\) \(\chi_{784}(747,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((687,197,689)\) → \((-1,i,e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 784 }(299, a) \) \(1\)\(1\)\(e\left(\frac{79}{84}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{31}{84}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{23}{42}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 784 }(299,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 784 }(299,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 784 }(299,·),\chi_{ 784 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 784 }(299,·)) \;\) at \(\; a,b = \) e.g. 1,2