Properties

Label 5915.48
Modulus $5915$
Conductor $5915$
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(5915, base_ring=CyclotomicField(156))
 
M = H._module
 
chi = DirichletCharacter(H, M([117,78,128]))
 
pari: [g,chi] = znchar(Mod(48,5915))
 

Basic properties

Modulus: \(5915\)
Conductor: \(5915\)
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 5915.gp

\(\chi_{5915}(48,\cdot)\) \(\chi_{5915}(237,\cdot)\) \(\chi_{5915}(328,\cdot)\) \(\chi_{5915}(412,\cdot)\) \(\chi_{5915}(503,\cdot)\) \(\chi_{5915}(692,\cdot)\) \(\chi_{5915}(783,\cdot)\) \(\chi_{5915}(958,\cdot)\) \(\chi_{5915}(1147,\cdot)\) \(\chi_{5915}(1238,\cdot)\) \(\chi_{5915}(1322,\cdot)\) \(\chi_{5915}(1413,\cdot)\) \(\chi_{5915}(1602,\cdot)\) \(\chi_{5915}(1693,\cdot)\) \(\chi_{5915}(1777,\cdot)\) \(\chi_{5915}(1868,\cdot)\) \(\chi_{5915}(2057,\cdot)\) \(\chi_{5915}(2148,\cdot)\) \(\chi_{5915}(2232,\cdot)\) \(\chi_{5915}(2323,\cdot)\) \(\chi_{5915}(2603,\cdot)\) \(\chi_{5915}(2687,\cdot)\) \(\chi_{5915}(2778,\cdot)\) \(\chi_{5915}(2967,\cdot)\) \(\chi_{5915}(3058,\cdot)\) \(\chi_{5915}(3142,\cdot)\) \(\chi_{5915}(3422,\cdot)\) \(\chi_{5915}(3513,\cdot)\) \(\chi_{5915}(3597,\cdot)\) \(\chi_{5915}(3688,\cdot)\) ...

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

\((2367,5071,1016)\) → \((-i,-1,e\left(\frac{32}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(16\)\(17\)
\( \chi_{ 5915 }(48, a) \) \(1\)\(1\)\(e\left(\frac{89}{156}\right)\)\(e\left(\frac{77}{156}\right)\)\(e\left(\frac{11}{78}\right)\)\(e\left(\frac{5}{78}\right)\)\(e\left(\frac{37}{52}\right)\)\(e\left(\frac{77}{78}\right)\)\(e\left(\frac{20}{39}\right)\)\(e\left(\frac{33}{52}\right)\)\(e\left(\frac{11}{39}\right)\)\(e\left(\frac{7}{156}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5915 }(48,a) \;\) at \(\;a = \) e.g. 2