Properties

Label 8112.67
Modulus $8112$
Conductor $2704$
Order $156$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8112, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,117,0,37]))
 
pari: [g,chi] = znchar(Mod(67,8112))
 

Basic properties

Modulus: \(8112\)
Conductor: \(2704\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2704}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8112.ga

\(\chi_{8112}(67,\cdot)\) \(\chi_{8112}(475,\cdot)\) \(\chi_{8112}(643,\cdot)\) \(\chi_{8112}(691,\cdot)\) \(\chi_{8112}(1051,\cdot)\) \(\chi_{8112}(1099,\cdot)\) \(\chi_{8112}(1267,\cdot)\) \(\chi_{8112}(1315,\cdot)\) \(\chi_{8112}(1675,\cdot)\) \(\chi_{8112}(1723,\cdot)\) \(\chi_{8112}(1891,\cdot)\) \(\chi_{8112}(2299,\cdot)\) \(\chi_{8112}(2515,\cdot)\) \(\chi_{8112}(2563,\cdot)\) \(\chi_{8112}(2923,\cdot)\) \(\chi_{8112}(2971,\cdot)\) \(\chi_{8112}(3139,\cdot)\) \(\chi_{8112}(3187,\cdot)\) \(\chi_{8112}(3547,\cdot)\) \(\chi_{8112}(3595,\cdot)\) \(\chi_{8112}(3763,\cdot)\) \(\chi_{8112}(3811,\cdot)\) \(\chi_{8112}(4171,\cdot)\) \(\chi_{8112}(4219,\cdot)\) \(\chi_{8112}(4387,\cdot)\) \(\chi_{8112}(4435,\cdot)\) \(\chi_{8112}(4795,\cdot)\) \(\chi_{8112}(4843,\cdot)\) \(\chi_{8112}(5011,\cdot)\) \(\chi_{8112}(5059,\cdot)\) ...

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: $\Q(\zeta_{156})$
Fixed field: Number field defined by a degree 156 polynomial (not computed)

Values on generators

\((5071,6085,2705,3889)\) → \((-1,-i,1,e\left(\frac{37}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 8112 }(67, a) \) \(1\)\(1\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{59}{156}\right)\)\(e\left(\frac{53}{78}\right)\)\(e\left(\frac{49}{78}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{115}{156}\right)\)\(e\left(\frac{25}{52}\right)\)\(e\left(\frac{41}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8112 }(67,a) \;\) at \(\;a = \) e.g. 2