Properties

Label 8041.597
Modulus $8041$
Conductor $8041$
Order $840$
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(840))
 
M = H._module
 
chi = DirichletCharacter(H, M([672,735,80]))
 
pari: [g,chi] = znchar(Mod(597,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(840\)
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.fy

\(\chi_{8041}(9,\cdot)\) \(\chi_{8041}(15,\cdot)\) \(\chi_{8041}(25,\cdot)\) \(\chi_{8041}(53,\cdot)\) \(\chi_{8041}(60,\cdot)\) \(\chi_{8041}(185,\cdot)\) \(\chi_{8041}(196,\cdot)\) \(\chi_{8041}(212,\cdot)\) \(\chi_{8041}(229,\cdot)\) \(\chi_{8041}(240,\cdot)\) \(\chi_{8041}(246,\cdot)\) \(\chi_{8041}(400,\cdot)\) \(\chi_{8041}(410,\cdot)\) \(\chi_{8041}(427,\cdot)\) \(\chi_{8041}(444,\cdot)\) \(\chi_{8041}(576,\cdot)\) \(\chi_{8041}(597,\cdot)\) \(\chi_{8041}(654,\cdot)\) \(\chi_{8041}(740,\cdot)\) \(\chi_{8041}(746,\cdot)\) \(\chi_{8041}(784,\cdot)\) \(\chi_{8041}(797,\cdot)\) \(\chi_{8041}(841,\cdot)\) \(\chi_{8041}(916,\cdot)\) \(\chi_{8041}(927,\cdot)\) \(\chi_{8041}(960,\cdot)\) \(\chi_{8041}(961,\cdot)\) \(\chi_{8041}(971,\cdot)\) \(\chi_{8041}(977,\cdot)\) \(\chi_{8041}(984,\cdot)\) ...

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: $\Q(\zeta_{840})$
Fixed field: Number field defined by a degree 840 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{7}{8}\right),e\left(\frac{2}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(597, a) \) \(1\)\(1\)\(e\left(\frac{87}{140}\right)\)\(e\left(\frac{311}{840}\right)\)\(e\left(\frac{17}{70}\right)\)\(e\left(\frac{803}{840}\right)\)\(e\left(\frac{119}{120}\right)\)\(e\left(\frac{67}{120}\right)\)\(e\left(\frac{121}{140}\right)\)\(e\left(\frac{311}{420}\right)\)\(e\left(\frac{97}{168}\right)\)\(e\left(\frac{103}{168}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(597,a) \;\) at \(\;a = \) e.g. 2