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Properties

Label 1560.47
Modulus $1560$
Conductor $780$
Order $4$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1560, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,0,2,1,1]))
 
Copy content pari:[g,chi] = znchar(Mod(47,1560))
 

Basic properties

Modulus: \(1560\)
Conductor: \(780\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(4\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{780}(47,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1560.bo

\(\chi_{1560}(47,\cdot)\) \(\chi_{1560}(863,\cdot)\)

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

Related number fields

Field of values: \(\mathbb{Q}(i)\)
Fixed field: 4.4.39546000.1

Values on generators

\((391,781,521,937,1081)\) → \((-1,1,-1,i,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1560 }(47, a) \) \(1\)\(1\)\(-1\)\(-i\)\(i\)\(i\)\(i\)\(1\)\(-i\)\(1\)\(-i\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1560 }(47,a) \;\) at \(\;a = \) e.g. 2