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Properties

Label 1680.1037
Modulus $1680$
Conductor $240$
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(1680, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,3,2,1,0]))
 
Copy content pari:[g,chi] = znchar(Mod(1037,1680))
 

Basic properties

Modulus: \(1680\)
Conductor: \(240\)
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 \(\chi_{240}(77,\cdot)\)
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 1680.cu

\(\chi_{1680}(533,\cdot)\) \(\chi_{1680}(1037,\cdot)\)

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: 4.4.2304000.2

Values on generators

\((1471,421,1121,337,241)\) → \((1,-i,-1,i,1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1680 }(1037, a) \) \(1\)\(1\)\(i\)\(1\)\(-i\)\(-i\)\(-i\)\(i\)\(1\)\(1\)\(1\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1680 }(1037,a) \;\) at \(\;a = \) e.g. 2