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Properties

Label 2040.13
Modulus $2040$
Conductor $680$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,2,0,3,1]))
 
Copy content pari:[g,chi] = znchar(Mod(13,2040))
 

Basic properties

Modulus: \(2040\)
Conductor: \(680\)
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 \(\chi_{680}(13,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 2040.bi

\(\chi_{2040}(13,\cdot)\) \(\chi_{2040}(157,\cdot)\)

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: 4.0.39304000.2

Values on generators

\((511,1021,1361,817,241)\) → \((1,-1,1,-i,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2040 }(13, a) \) \(-1\)\(1\)\(-1\)\(i\)\(-i\)\(-1\)\(1\)\(i\)\(i\)\(-1\)\(-i\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2040 }(13,a) \;\) at \(\;a = \) e.g. 2