Properties

Label 8041.8025
Modulus $8041$
Conductor $473$
Order $70$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,0,5]))
 
pari: [g,chi] = znchar(Mod(8025,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(457,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.dt

\(\chi_{8041}(409,\cdot)\) \(\chi_{8041}(426,\cdot)\) \(\chi_{8041}(1140,\cdot)\) \(\chi_{8041}(1157,\cdot)\) \(\chi_{8041}(1888,\cdot)\) \(\chi_{8041}(2109,\cdot)\) \(\chi_{8041}(2483,\cdot)\) \(\chi_{8041}(2602,\cdot)\) \(\chi_{8041}(2840,\cdot)\) \(\chi_{8041}(3214,\cdot)\) \(\chi_{8041}(3350,\cdot)\) \(\chi_{8041}(3571,\cdot)\) \(\chi_{8041}(3792,\cdot)\) \(\chi_{8041}(3945,\cdot)\) \(\chi_{8041}(4523,\cdot)\) \(\chi_{8041}(5033,\cdot)\) \(\chi_{8041}(5101,\cdot)\) \(\chi_{8041}(5254,\cdot)\) \(\chi_{8041}(5407,\cdot)\) \(\chi_{8041}(5832,\cdot)\) \(\chi_{8041}(6563,\cdot)\) \(\chi_{8041}(6716,\cdot)\) \(\chi_{8041}(7719,\cdot)\) \(\chi_{8041}(8025,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{9}{10}\right),1,e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(8025, a) \) \(1\)\(1\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{19}{70}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{13}{14}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(8025,a) \;\) at \(\;a = \) e.g. 2