Properties

Label 392.361
Modulus $392$
Conductor $7$
Order $3$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,4]))
 
pari: [g,chi] = znchar(Mod(361,392))
 

Basic properties

Modulus: \(392\)
Conductor: \(7\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{7}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 392.i

\(\chi_{392}(177,\cdot)\) \(\chi_{392}(361,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: \(\Q(\zeta_{7})^+\)

Values on generators

\((295,197,297)\) → \((1,1,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 392 }(361, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 392 }(361,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 392 }(361,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 392 }(361,·),\chi_{ 392 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 392 }(361,·)) \;\) at \(\; a,b = \) e.g. 1,2