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Properties

Label 2340.863
Modulus $2340$
Conductor $780$
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(2340, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([2,2,3,3]))
 
Copy content pari:[g,chi] = znchar(Mod(863,2340))
 

Basic properties

Modulus: \(2340\)
Conductor: \(780\)
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 \(\chi_{780}(83,\cdot)\)
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 2340.x

\(\chi_{2340}(827,\cdot)\) \(\chi_{2340}(863,\cdot)\)

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: 4.4.39546000.1

Values on generators

\((1171,2081,937,1081)\) → \((-1,-1,-i,-i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2340 }(863, a) \) \(1\)\(1\)\(-1\)\(i\)\(-i\)\(-i\)\(-i\)\(1\)\(i\)\(1\)\(i\)\(i\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2340 }(863,a) \;\) at \(\;a = \) e.g. 2