Properties

Label 8041.412
Modulus $8041$
Conductor $8041$
Order $420$
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(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([168,315,80]))
 
pari: [g,chi] = znchar(Mod(412,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
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.fo

\(\chi_{8041}(38,\cdot)\) \(\chi_{8041}(81,\cdot)\) \(\chi_{8041}(225,\cdot)\) \(\chi_{8041}(268,\cdot)\) \(\chi_{8041}(361,\cdot)\) \(\chi_{8041}(412,\cdot)\) \(\chi_{8041}(455,\cdot)\) \(\chi_{8041}(531,\cdot)\) \(\chi_{8041}(599,\cdot)\) \(\chi_{8041}(625,\cdot)\) \(\chi_{8041}(642,\cdot)\) \(\chi_{8041}(676,\cdot)\) \(\chi_{8041}(812,\cdot)\) \(\chi_{8041}(999,\cdot)\) \(\chi_{8041}(1092,\cdot)\) \(\chi_{8041}(1186,\cdot)\) \(\chi_{8041}(1313,\cdot)\) \(\chi_{8041}(1347,\cdot)\) \(\chi_{8041}(1356,\cdot)\) \(\chi_{8041}(1373,\cdot)\) \(\chi_{8041}(1390,\cdot)\) \(\chi_{8041}(1500,\cdot)\) \(\chi_{8041}(1543,\cdot)\) \(\chi_{8041}(1687,\cdot)\) \(\chi_{8041}(1730,\cdot)\) \(\chi_{8041}(1874,\cdot)\) \(\chi_{8041}(1917,\cdot)\) \(\chi_{8041}(2044,\cdot)\) \(\chi_{8041}(2061,\cdot)\) \(\chi_{8041}(2095,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(412, a) \) \(1\)\(1\)\(e\left(\frac{3}{70}\right)\)\(e\left(\frac{59}{420}\right)\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{47}{420}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{9}{70}\right)\)\(e\left(\frac{59}{210}\right)\)\(e\left(\frac{13}{84}\right)\)\(e\left(\frac{19}{84}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(412,a) \;\) at \(\;a = \) e.g. 2