Properties

Label 8027.3165
Modulus $8027$
Conductor $8027$
Order $132$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([126,11]))
 
pari: [g,chi] = znchar(Mod(3165,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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 8027.be

\(\chi_{8027}(189,\cdot)\) \(\chi_{8027}(373,\cdot)\) \(\chi_{8027}(674,\cdot)\) \(\chi_{8027}(858,\cdot)\) \(\chi_{8027}(1023,\cdot)\) \(\chi_{8027}(1207,\cdot)\) \(\chi_{8027}(1236,\cdot)\) \(\chi_{8027}(1372,\cdot)\) \(\chi_{8027}(1420,\cdot)\) \(\chi_{8027}(1556,\cdot)\) \(\chi_{8027}(1585,\cdot)\) \(\chi_{8027}(1721,\cdot)\) \(\chi_{8027}(1769,\cdot)\) \(\chi_{8027}(1905,\cdot)\) \(\chi_{8027}(2632,\cdot)\) \(\chi_{8027}(2816,\cdot)\) \(\chi_{8027}(2981,\cdot)\) \(\chi_{8027}(3165,\cdot)\) \(\chi_{8027}(3815,\cdot)\) \(\chi_{8027}(3999,\cdot)\) \(\chi_{8027}(4377,\cdot)\) \(\chi_{8027}(4513,\cdot)\) \(\chi_{8027}(4561,\cdot)\) \(\chi_{8027}(4697,\cdot)\) \(\chi_{8027}(4726,\cdot)\) \(\chi_{8027}(4910,\cdot)\) \(\chi_{8027}(5075,\cdot)\) \(\chi_{8027}(5259,\cdot)\) \(\chi_{8027}(5424,\cdot)\) \(\chi_{8027}(5560,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((350,5935)\) → \((e\left(\frac{21}{22}\right),e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(3165, a) \) \(1\)\(1\)\(e\left(\frac{131}{132}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{95}{132}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{15}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(3165,a) \;\) at \(\;a = \) e.g. 2