Properties

Label 91.8
Modulus $91$
Conductor $13$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,1]))
 
pari: [g,chi] = znchar(Mod(8,91))
 

Basic properties

Modulus: \(91\)
Conductor: \(13\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{13}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 91.j

\(\chi_{91}(8,\cdot)\) \(\chi_{91}(57,\cdot)\)

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: \(\mathbb{Q}(i)\)
Fixed field: 4.0.2197.1

Values on generators

\((66,15)\) → \((1,i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 91 }(8, a) \) \(-1\)\(1\)\(i\)\(1\)\(-1\)\(i\)\(i\)\(-i\)\(1\)\(-1\)\(-i\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 91 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 91 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 91 }(8,·),\chi_{ 91 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 91 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2