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Properties

Label 1248.961
Modulus $1248$
Conductor $13$
Order $2$
Real yes
Primitive no
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1248, base_ring=CyclotomicField(2)) M = H._module chi = DirichletCharacter(H, M([0,0,0,1]))
 
Copy content pari:[g,chi] = znchar(Mod(961,1248))
 

Basic properties

Modulus: \(1248\)
Conductor: \(13\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(2\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{13}(12,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1248.c

\(\chi_{1248}(961,\cdot)\)

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: \(\Q\)
Fixed field: \(\Q(\sqrt{13}) \)

Values on generators

\((703,1093,833,769)\) → \((1,1,1,-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1248 }(961, a) \) \(1\)\(1\)\(-1\)\(-1\)\(-1\)\(1\)\(-1\)\(1\)\(1\)\(1\)\(-1\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1248 }(961,a) \;\) at \(\;a = \) e.g. 2