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Properties

Label 8352.7001
Modulus $8352$
Conductor $1392$
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(8352, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,1,2,1]))
 
Copy content pari:[g,chi] = znchar(Mod(7001,8352))
 

Basic properties

Modulus: \(8352\)
Conductor: \(1392\)
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 \(\chi_{1392}(389,\cdot)\)
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 8352.bc

\(\chi_{8352}(6281,\cdot)\) \(\chi_{8352}(7001,\cdot)\)

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: 4.4.449538048.1

Values on generators

\((1567,5221,929,4033)\) → \((1,i,-1,i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(31\)\(35\)
\( \chi_{ 8352 }(7001, a) \) \(1\)\(1\)\(i\)\(-1\)\(1\)\(i\)\(-i\)\(1\)\(1\)\(-1\)\(i\)\(-i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 8352 }(7001,a) \;\) at \(\;a = \) e.g. 2