Properties

Label 392.269
Modulus $392$
Conductor $392$
Order $42$
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(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,37]))
 
pari: [g,chi] = znchar(Mod(269,392))
 

Basic properties

Modulus: \(392\)
Conductor: \(392\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.bf

\(\chi_{392}(5,\cdot)\) \(\chi_{392}(45,\cdot)\) \(\chi_{392}(61,\cdot)\) \(\chi_{392}(101,\cdot)\) \(\chi_{392}(157,\cdot)\) \(\chi_{392}(173,\cdot)\) \(\chi_{392}(213,\cdot)\) \(\chi_{392}(229,\cdot)\) \(\chi_{392}(269,\cdot)\) \(\chi_{392}(285,\cdot)\) \(\chi_{392}(341,\cdot)\) \(\chi_{392}(381,\cdot)\)

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: 42.0.1090030896264192289800449659845679818091197961133776603876122561317234873686091104256.1

Values on generators

\((295,197,297)\) → \((1,-1,e\left(\frac{37}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 392 }(269, a) \) \(-1\)\(1\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{2}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 392 }(269,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 392 }(269,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 392 }(269,·),\chi_{ 392 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 392 }(269,·)) \;\) at \(\; a,b = \) e.g. 1,2