Properties

Label 784.39
Modulus $784$
Conductor $392$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,21,34]))
 
pari: [g,chi] = znchar(Mod(39,784))
 

Basic properties

Modulus: \(784\)
Conductor: \(392\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{392}(235,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 784.bq

\(\chi_{784}(23,\cdot)\) \(\chi_{784}(39,\cdot)\) \(\chi_{784}(135,\cdot)\) \(\chi_{784}(151,\cdot)\) \(\chi_{784}(247,\cdot)\) \(\chi_{784}(359,\cdot)\) \(\chi_{784}(375,\cdot)\) \(\chi_{784}(487,\cdot)\) \(\chi_{784}(583,\cdot)\) \(\chi_{784}(599,\cdot)\) \(\chi_{784}(695,\cdot)\) \(\chi_{784}(711,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.155718699466313184257207094263668545441599708733396657696588937331033553383727300608.1

Values on generators

\((687,197,689)\) → \((-1,-1,e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 784 }(39, a) \) \(-1\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{20}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 784 }(39,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 784 }(39,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 784 }(39,·),\chi_{ 784 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 784 }(39,·)) \;\) at \(\; a,b = \) e.g. 1,2