Properties

Label 85.78
Modulus $85$
Conductor $85$
Order $16$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(85, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,3]))
 
pari: [g,chi] = znchar(Mod(78,85))
 

Basic properties

Modulus: \(85\)
Conductor: \(85\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 85.r

\(\chi_{85}(12,\cdot)\) \(\chi_{85}(22,\cdot)\) \(\chi_{85}(23,\cdot)\) \(\chi_{85}(28,\cdot)\) \(\chi_{85}(37,\cdot)\) \(\chi_{85}(58,\cdot)\) \(\chi_{85}(78,\cdot)\) \(\chi_{85}(82,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.16.698833752810013621337890625.1

Values on generators

\((52,71)\) → \((-i,e\left(\frac{3}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 85 }(78, a) \) \(1\)\(1\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{7}{16}\right)\)\(-i\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 85 }(78,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 85 }(78,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 85 }(78,·),\chi_{ 85 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 85 }(78,·)) \;\) at \(\; a,b = \) e.g. 1,2