Properties

Label 6760.37
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([0,78,39,151]))
 
pari: [g,chi] = znchar(Mod(37,6760))
 

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.fy

\(\chi_{6760}(37,\cdot)\) \(\chi_{6760}(93,\cdot)\) \(\chi_{6760}(253,\cdot)\) \(\chi_{6760}(397,\cdot)\) \(\chi_{6760}(557,\cdot)\) \(\chi_{6760}(613,\cdot)\) \(\chi_{6760}(773,\cdot)\) \(\chi_{6760}(917,\cdot)\) \(\chi_{6760}(1077,\cdot)\) \(\chi_{6760}(1133,\cdot)\) \(\chi_{6760}(1293,\cdot)\) \(\chi_{6760}(1437,\cdot)\) \(\chi_{6760}(1597,\cdot)\) \(\chi_{6760}(1653,\cdot)\) \(\chi_{6760}(1813,\cdot)\) \(\chi_{6760}(1957,\cdot)\) \(\chi_{6760}(2173,\cdot)\) \(\chi_{6760}(2333,\cdot)\) \(\chi_{6760}(2477,\cdot)\) \(\chi_{6760}(2637,\cdot)\) \(\chi_{6760}(2693,\cdot)\) \(\chi_{6760}(2853,\cdot)\) \(\chi_{6760}(2997,\cdot)\) \(\chi_{6760}(3157,\cdot)\) \(\chi_{6760}(3213,\cdot)\) \(\chi_{6760}(3373,\cdot)\) \(\chi_{6760}(3517,\cdot)\) \(\chi_{6760}(3677,\cdot)\) \(\chi_{6760}(3733,\cdot)\) \(\chi_{6760}(3893,\cdot)\) ...

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

\((5071,3381,4057,5241)\) → \((1,-1,i,e\left(\frac{151}{156}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 6760 }(37, a) \) \(1\)\(1\)\(e\left(\frac{43}{156}\right)\)\(e\left(\frac{32}{39}\right)\)\(e\left(\frac{43}{78}\right)\)\(e\left(\frac{31}{156}\right)\)\(e\left(\frac{89}{156}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{52}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{43}{52}\right)\)\(e\left(\frac{28}{39}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6760 }(37,a) \;\) at \(\;a = \) e.g. 2