Properties

Label 8041.6614
Modulus $8041$
Conductor $473$
Order $35$
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([56,0,30]))
 
pari: [g,chi] = znchar(Mod(6614,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(465,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8041.cr

\(\chi_{8041}(256,\cdot)\) \(\chi_{8041}(477,\cdot)\) \(\chi_{8041}(1208,\cdot)\) \(\chi_{8041}(1225,\cdot)\) \(\chi_{8041}(1939,\cdot)\) \(\chi_{8041}(1956,\cdot)\) \(\chi_{8041}(2381,\cdot)\) \(\chi_{8041}(2687,\cdot)\) \(\chi_{8041}(3690,\cdot)\) \(\chi_{8041}(3843,\cdot)\) \(\chi_{8041}(4574,\cdot)\) \(\chi_{8041}(4999,\cdot)\) \(\chi_{8041}(5152,\cdot)\) \(\chi_{8041}(5305,\cdot)\) \(\chi_{8041}(5373,\cdot)\) \(\chi_{8041}(5883,\cdot)\) \(\chi_{8041}(6461,\cdot)\) \(\chi_{8041}(6614,\cdot)\) \(\chi_{8041}(6835,\cdot)\) \(\chi_{8041}(7056,\cdot)\) \(\chi_{8041}(7192,\cdot)\) \(\chi_{8041}(7566,\cdot)\) \(\chi_{8041}(7804,\cdot)\) \(\chi_{8041}(7923,\cdot)\)

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 35 polynomial

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{4}{5}\right),1,e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(6614, a) \) \(1\)\(1\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{32}{35}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(6614,a) \;\) at \(\;a = \) e.g. 2