Properties

Label 3120.1249
Modulus $3120$
Conductor $5$
Order $2$
Real yes
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3120, base_ring=CyclotomicField(2))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,0,1,0]))
 
pari: [g,chi] = znchar(Mod(1249,3120))
 

Basic properties

Modulus: \(3120\)
Conductor: \(5\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{5}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3120.l

\(\chi_{3120}(1249,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{5}) \)

Values on generators

\((1951,2341,2081,2497,2641)\) → \((1,1,1,-1,1)\)

Values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3120 }(1249, a) \) \(1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3120 }(1249,a) \;\) at \(\;a = \) e.g. 2