Properties

Label 6760.119
Modulus $6760$
Conductor $3380$
Order $156$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6760, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([78,0,78,97]))
 
pari: [g,chi] = znchar(Mod(119,6760))
 

Basic properties

Modulus: \(6760\)
Conductor: \(3380\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{3380}(119,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6760.gd

\(\chi_{6760}(119,\cdot)\) \(\chi_{6760}(279,\cdot)\) \(\chi_{6760}(479,\cdot)\) \(\chi_{6760}(639,\cdot)\) \(\chi_{6760}(799,\cdot)\) \(\chi_{6760}(839,\cdot)\) \(\chi_{6760}(999,\cdot)\) \(\chi_{6760}(1159,\cdot)\) \(\chi_{6760}(1319,\cdot)\) \(\chi_{6760}(1359,\cdot)\) \(\chi_{6760}(1519,\cdot)\) \(\chi_{6760}(1679,\cdot)\) \(\chi_{6760}(1839,\cdot)\) \(\chi_{6760}(1879,\cdot)\) \(\chi_{6760}(2039,\cdot)\) \(\chi_{6760}(2199,\cdot)\) \(\chi_{6760}(2359,\cdot)\) \(\chi_{6760}(2399,\cdot)\) \(\chi_{6760}(2559,\cdot)\) \(\chi_{6760}(2719,\cdot)\) \(\chi_{6760}(2879,\cdot)\) \(\chi_{6760}(2919,\cdot)\) \(\chi_{6760}(3079,\cdot)\) \(\chi_{6760}(3239,\cdot)\) \(\chi_{6760}(3439,\cdot)\) \(\chi_{6760}(3599,\cdot)\) \(\chi_{6760}(3759,\cdot)\) \(\chi_{6760}(3919,\cdot)\) \(\chi_{6760}(3959,\cdot)\) \(\chi_{6760}(4119,\cdot)\) ...

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

\((5071,3381,4057,5241)\) → \((-1,1,-1,e\left(\frac{97}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 6760 }(119, a) \) \(1\)\(1\)\(e\left(\frac{4}{39}\right)\)\(e\left(\frac{83}{156}\right)\)\(e\left(\frac{8}{39}\right)\)\(e\left(\frac{85}{156}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{4}{13}\right)\)\(e\left(\frac{34}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6760 }(119,a) \;\) at \(\;a = \) e.g. 2