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Properties

Label 2720.1857
Modulus $2720$
Conductor $85$
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(2720, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,0,1,3]))
 
Copy content pari:[g,chi] = znchar(Mod(1857,2720))
 

Basic properties

Modulus: \(2720\)
Conductor: \(85\)
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 \(\chi_{85}(72,\cdot)\)
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 2720.r

\(\chi_{2720}(353,\cdot)\) \(\chi_{2720}(1857,\cdot)\)

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: 4.0.614125.2

Values on generators

\((511,1701,2177,1601)\) → \((1,1,i,-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 2720 }(1857, a) \) \(-1\)\(1\)\(-1\)\(-1\)\(1\)\(i\)\(-i\)\(1\)\(1\)\(1\)\(-1\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2720 }(1857,a) \;\) at \(\;a = \) e.g. 2