Properties

Label 336.85
Modulus $336$
Conductor $16$
Order $4$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1,0,0]))
 
pari: [g,chi] = znchar(Mod(85,336))
 

Basic properties

Modulus: \(336\)
Conductor: \(16\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{16}(5,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 336.w

\(\chi_{336}(85,\cdot)\) \(\chi_{336}(253,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: \(\Q(\zeta_{16})^+\)

Values on generators

\((127,85,113,241)\) → \((1,i,1,1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 336 }(85, a) \) \(1\)\(1\)\(i\)\(i\)\(-i\)\(1\)\(-i\)\(-1\)\(-1\)\(-i\)\(1\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 336 }(85,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 336 }(85,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 336 }(85,·),\chi_{ 336 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 336 }(85,·)) \;\) at \(\; a,b = \) e.g. 1,2