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Properties

Label 4056.3281
Modulus $4056$
Conductor $39$
Order $4$
Real no
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(4056, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,0,2,3]))
 
Copy content pari:[g,chi] = znchar(Mod(3281,4056))
 

Basic properties

Modulus: \(4056\)
Conductor: \(39\)
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 \(\chi_{39}(5,\cdot)\)
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 4056.x

\(\chi_{4056}(2465,\cdot)\) \(\chi_{4056}(3281,\cdot)\)

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: 4.4.19773.1

Values on generators

\((1015,2029,2705,3889)\) → \((1,1,-1,-i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 4056 }(3281, a) \) \(1\)\(1\)\(i\)\(i\)\(-i\)\(1\)\(-i\)\(1\)\(-1\)\(-1\)\(-i\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 4056 }(3281,a) \;\) at \(\;a = \) e.g. 2