Properties

Label 8023.91
Modulus $8023$
Conductor $8023$
Order $56$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8023, base_ring=CyclotomicField(56))
 
M = H._module
 
chi = DirichletCharacter(H, M([32,15]))
 
pari: [g,chi] = znchar(Mod(91,8023))
 

Basic properties

Modulus: \(8023\)
Conductor: \(8023\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(56\)
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 8023.de

\(\chi_{8023}(91,\cdot)\) \(\chi_{8023}(400,\cdot)\) \(\chi_{8023}(529,\cdot)\) \(\chi_{8023}(534,\cdot)\) \(\chi_{8023}(616,\cdot)\) \(\chi_{8023}(1381,\cdot)\) \(\chi_{8023}(2814,\cdot)\) \(\chi_{8023}(3073,\cdot)\) \(\chi_{8023}(3286,\cdot)\) \(\chi_{8023}(3516,\cdot)\) \(\chi_{8023}(3942,\cdot)\) \(\chi_{8023}(4305,\cdot)\) \(\chi_{8023}(4787,\cdot)\) \(\chi_{8023}(5286,\cdot)\) \(\chi_{8023}(5373,\cdot)\) \(\chi_{8023}(6138,\cdot)\) \(\chi_{8023}(6278,\cdot)\) \(\chi_{8023}(6491,\cdot)\) \(\chi_{8023}(6580,\cdot)\) \(\chi_{8023}(6719,\cdot)\) \(\chi_{8023}(6924,\cdot)\) \(\chi_{8023}(7432,\cdot)\) \(\chi_{8023}(7901,\cdot)\) \(\chi_{8023}(7982,\cdot)\)

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: $\Q(\zeta_{56})$
Fixed field: Number field defined by a degree 56 polynomial

Values on generators

\((6894,3054)\) → \((e\left(\frac{4}{7}\right),e\left(\frac{15}{56}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8023 }(91, a) \) \(1\)\(1\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{13}{56}\right)\)\(e\left(\frac{43}{56}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{15}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8023 }(91,a) \;\) at \(\;a = \) e.g. 2