Properties

Label 8021.113
Modulus $8021$
Conductor $8021$
Order $33$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,54]))
 
pari: [g,chi] = znchar(Mod(113,8021))
 

Basic properties

Modulus: \(8021\)
Conductor: \(8021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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.br

\(\chi_{8021}(113,\cdot)\) \(\chi_{8021}(1576,\cdot)\) \(\chi_{8021}(2193,\cdot)\) \(\chi_{8021}(2499,\cdot)\) \(\chi_{8021}(3116,\cdot)\) \(\chi_{8021}(3877,\cdot)\) \(\chi_{8021}(4046,\cdot)\) \(\chi_{8021}(4494,\cdot)\) \(\chi_{8021}(4663,\cdot)\) \(\chi_{8021}(4670,\cdot)\) \(\chi_{8021}(4748,\cdot)\) \(\chi_{8021}(5287,\cdot)\) \(\chi_{8021}(5365,\cdot)\) \(\chi_{8021}(6659,\cdot)\) \(\chi_{8021}(7179,\cdot)\) \(\chi_{8021}(7205,\cdot)\) \(\chi_{8021}(7276,\cdot)\) \(\chi_{8021}(7517,\cdot)\) \(\chi_{8021}(7796,\cdot)\) \(\chi_{8021}(7822,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\((6788,2471)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8021 }(113, a) \) \(1\)\(1\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{33}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{31}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8021 }(113,a) \;\) at \(\;a = \) e.g. 2