Properties

Label 6760.197
Modulus $6760$
Conductor $6760$
Order $156$
Real no
Primitive yes
Minimal yes
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([0,78,39,109]))
 
pari: [g,chi] = znchar(Mod(197,6760))
 

Basic properties

Modulus: \(6760\)
Conductor: \(6760\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(156\)
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 6760.fn

\(\chi_{6760}(197,\cdot)\) \(\chi_{6760}(293,\cdot)\) \(\chi_{6760}(453,\cdot)\) \(\chi_{6760}(717,\cdot)\) \(\chi_{6760}(813,\cdot)\) \(\chi_{6760}(877,\cdot)\) \(\chi_{6760}(973,\cdot)\) \(\chi_{6760}(1237,\cdot)\) \(\chi_{6760}(1397,\cdot)\) \(\chi_{6760}(1493,\cdot)\) \(\chi_{6760}(1757,\cdot)\) \(\chi_{6760}(1853,\cdot)\) \(\chi_{6760}(1917,\cdot)\) \(\chi_{6760}(2013,\cdot)\) \(\chi_{6760}(2373,\cdot)\) \(\chi_{6760}(2437,\cdot)\) \(\chi_{6760}(2533,\cdot)\) \(\chi_{6760}(2797,\cdot)\) \(\chi_{6760}(2893,\cdot)\) \(\chi_{6760}(2957,\cdot)\) \(\chi_{6760}(3053,\cdot)\) \(\chi_{6760}(3317,\cdot)\) \(\chi_{6760}(3413,\cdot)\) \(\chi_{6760}(3477,\cdot)\) \(\chi_{6760}(3573,\cdot)\) \(\chi_{6760}(3837,\cdot)\) \(\chi_{6760}(3933,\cdot)\) \(\chi_{6760}(3997,\cdot)\) \(\chi_{6760}(4093,\cdot)\) \(\chi_{6760}(4357,\cdot)\) ...

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

\((5071,3381,4057,5241)\) → \((1,-1,i,e\left(\frac{109}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 6760 }(197, a) \) \(1\)\(1\)\(e\left(\frac{139}{156}\right)\)\(e\left(\frac{1}{78}\right)\)\(e\left(\frac{61}{78}\right)\)\(e\left(\frac{73}{156}\right)\)\(e\left(\frac{41}{156}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{47}{52}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{35}{52}\right)\)\(e\left(\frac{37}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6760 }(197,a) \;\) at \(\;a = \) e.g. 2