Properties

Label 8041.5252
Modulus $8041$
Conductor $8041$
Order $30$
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(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,15,20]))
 
pari: [g,chi] = znchar(Mod(5252,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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.ck

\(\chi_{8041}(135,\cdot)\) \(\chi_{8041}(509,\cdot)\) \(\chi_{8041}(5252,\cdot)\) \(\chi_{8041}(5626,\cdot)\) \(\chi_{8041}(6714,\cdot)\) \(\chi_{8041}(7088,\cdot)\) \(\chi_{8041}(7445,\cdot)\) \(\chi_{8041}(7819,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{2}{5}\right),-1,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(5252, a) \) \(1\)\(1\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(5252,a) \;\) at \(\;a = \) e.g. 2