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Properties

Label 5712.13
Modulus $5712$
Conductor $1904$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5712, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,3,0,2,1]))
 
Copy content pari:[g,chi] = znchar(Mod(13,5712))
 

Basic properties

Modulus: \(5712\)
Conductor: \(1904\)
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 \(\chi_{1904}(13,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 5712.bi

\(\chi_{5712}(13,\cdot)\) \(\chi_{5712}(2197,\cdot)\)

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: 4.0.493029376.3

Values on generators

\((2143,1429,3809,3265,2689)\) → \((1,-i,1,-1,i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 5712 }(13, a) \) \(-1\)\(1\)\(-1\)\(-1\)\(-i\)\(i\)\(i\)\(1\)\(-1\)\(-i\)\(1\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 5712 }(13,a) \;\) at \(\;a = \) e.g. 2