Properties

Label 8041.7870
Modulus $8041$
Conductor $187$
Order $10$
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(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,5,0]))
 
pari: [g,chi] = znchar(Mod(7870,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(187\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{187}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.bc

\(\chi_{8041}(1291,\cdot)\) \(\chi_{8041}(2022,\cdot)\) \(\chi_{8041}(2753,\cdot)\) \(\chi_{8041}(7870,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.304358957700017.1

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{2}{5}\right),-1,1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(7870, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(-1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(7870,a) \;\) at \(\;a = \) e.g. 2