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Properties

Label 2312.251
Modulus $2312$
Conductor $136$
Order $4$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2312, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,2,1]))
 
Copy content pari:[g,chi] = znchar(Mod(251,2312))
 

Basic properties

Modulus: \(2312\)
Conductor: \(136\)
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 \(\chi_{136}(115,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 2312.j

\(\chi_{2312}(251,\cdot)\) \(\chi_{2312}(1483,\cdot)\)

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: 4.0.314432.2

Values on generators

\((1735,1157,1737)\) → \((-1,-1,i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(19\)\(21\)\(23\)
\( \chi_{ 2312 }(251, a) \) \(-1\)\(1\)\(i\)\(-i\)\(i\)\(-1\)\(-i\)\(-1\)\(1\)\(-1\)\(-1\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2312 }(251,a) \;\) at \(\;a = \) e.g. 2