Properties

Label 8041.4101
Modulus $8041$
Conductor $8041$
Order $140$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(140))
 
M = H._module
 
chi = DirichletCharacter(H, M([84,105,80]))
 
pari: [g,chi] = znchar(Mod(4101,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
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 8041.eo

\(\chi_{8041}(4,\cdot)\) \(\chi_{8041}(47,\cdot)\) \(\chi_{8041}(64,\cdot)\) \(\chi_{8041}(489,\cdot)\) \(\chi_{8041}(735,\cdot)\) \(\chi_{8041}(752,\cdot)\) \(\chi_{8041}(795,\cdot)\) \(\chi_{8041}(1466,\cdot)\) \(\chi_{8041}(1483,\cdot)\) \(\chi_{8041}(1798,\cdot)\) \(\chi_{8041}(1908,\cdot)\) \(\chi_{8041}(1951,\cdot)\) \(\chi_{8041}(2214,\cdot)\) \(\chi_{8041}(2682,\cdot)\) \(\chi_{8041}(3107,\cdot)\) \(\chi_{8041}(3217,\cdot)\) \(\chi_{8041}(3260,\cdot)\) \(\chi_{8041}(3370,\cdot)\) \(\chi_{8041}(3413,\cdot)\) \(\chi_{8041}(3481,\cdot)\) \(\chi_{8041}(3991,\cdot)\) \(\chi_{8041}(4101,\cdot)\) \(\chi_{8041}(4526,\cdot)\) \(\chi_{8041}(4569,\cdot)\) \(\chi_{8041}(4679,\cdot)\) \(\chi_{8041}(4722,\cdot)\) \(\chi_{8041}(4832,\cdot)\) \(\chi_{8041}(4900,\cdot)\) \(\chi_{8041}(4943,\cdot)\) \(\chi_{8041}(5164,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{3}{5}\right),-i,e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(4101, a) \) \(1\)\(1\)\(e\left(\frac{37}{70}\right)\)\(e\left(\frac{17}{140}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{61}{140}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{5}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(4101,a) \;\) at \(\;a = \) e.g. 2