Properties

Label 392.323
Modulus $392$
Conductor $392$
Order $14$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,7,6]))
 
pari: [g,chi] = znchar(Mod(323,392))
 

Basic properties

Modulus: \(392\)
Conductor: \(392\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 392.s

\(\chi_{392}(43,\cdot)\) \(\chi_{392}(155,\cdot)\) \(\chi_{392}(211,\cdot)\) \(\chi_{392}(267,\cdot)\) \(\chi_{392}(323,\cdot)\) \(\chi_{392}(379,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: Number field defined by a degree 14 polynomial

Values on generators

\((295,197,297)\) → \((-1,-1,e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 392 }(323, a) \) \(-1\)\(1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(1\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{6}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 392 }(323,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 392 }(323,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 392 }(323,·),\chi_{ 392 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 392 }(323,·)) \;\) at \(\; a,b = \) e.g. 1,2