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Properties

Label 2040.127
Modulus $2040$
Conductor $340$
Order $8$
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(2040, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([4,0,0,2,5]))
 
Copy content pari:[g,chi] = znchar(Mod(127,2040))
 

Basic properties

Modulus: \(2040\)
Conductor: \(340\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{340}(127,\cdot)\)
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 2040.dn

\(\chi_{2040}(127,\cdot)\) \(\chi_{2040}(223,\cdot)\) \(\chi_{2040}(247,\cdot)\) \(\chi_{2040}(1783,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.1641354692000000.2

Values on generators

\((511,1021,1361,817,241)\) → \((-1,1,1,i,e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2040 }(127, a) \) \(1\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(i\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2040 }(127,a) \;\) at \(\;a = \) e.g. 2