Properties

Label 8021.51
Modulus $8021$
Conductor $8021$
Order $154$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8021, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,97]))
 
pari: [g,chi] = znchar(Mod(51,8021))
 

Basic properties

Modulus: \(8021\)
Conductor: \(8021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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 8021.cz

\(\chi_{8021}(51,\cdot)\) \(\chi_{8021}(259,\cdot)\) \(\chi_{8021}(285,\cdot)\) \(\chi_{8021}(324,\cdot)\) \(\chi_{8021}(350,\cdot)\) \(\chi_{8021}(493,\cdot)\) \(\chi_{8021}(519,\cdot)\) \(\chi_{8021}(649,\cdot)\) \(\chi_{8021}(688,\cdot)\) \(\chi_{8021}(1013,\cdot)\) \(\chi_{8021}(1689,\cdot)\) \(\chi_{8021}(1936,\cdot)\) \(\chi_{8021}(2092,\cdot)\) \(\chi_{8021}(2391,\cdot)\) \(\chi_{8021}(2404,\cdot)\) \(\chi_{8021}(2664,\cdot)\) \(\chi_{8021}(2716,\cdot)\) \(\chi_{8021}(2755,\cdot)\) \(\chi_{8021}(2833,\cdot)\) \(\chi_{8021}(3002,\cdot)\) \(\chi_{8021}(3093,\cdot)\) \(\chi_{8021}(3132,\cdot)\) \(\chi_{8021}(3184,\cdot)\) \(\chi_{8021}(3353,\cdot)\) \(\chi_{8021}(3600,\cdot)\) \(\chi_{8021}(3639,\cdot)\) \(\chi_{8021}(3704,\cdot)\) \(\chi_{8021}(3769,\cdot)\) \(\chi_{8021}(3873,\cdot)\) \(\chi_{8021}(3912,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((6788,2471)\) → \((-1,e\left(\frac{97}{154}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8021 }(51, a) \) \(1\)\(1\)\(e\left(\frac{13}{154}\right)\)\(e\left(\frac{97}{154}\right)\)\(e\left(\frac{13}{77}\right)\)\(e\left(\frac{53}{77}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{145}{154}\right)\)\(e\left(\frac{39}{154}\right)\)\(e\left(\frac{20}{77}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{47}{154}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8021 }(51,a) \;\) at \(\;a = \) e.g. 2