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Properties

Label 2160.109
Modulus $2160$
Conductor $80$
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(2160, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,3,0,2]))
 
Copy content pari:[g,chi] = znchar(Mod(109,2160))
 

Basic properties

Modulus: \(2160\)
Conductor: \(80\)
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 \(\chi_{80}(29,\cdot)\)
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 2160.bm

\(\chi_{2160}(109,\cdot)\) \(\chi_{2160}(1189,\cdot)\)

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: 4.4.51200.1

Values on generators

\((271,1621,2081,1297)\) → \((1,-i,1,-1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 2160 }(109, a) \) \(1\)\(1\)\(1\)\(-i\)\(-i\)\(-1\)\(i\)\(1\)\(i\)\(1\)\(i\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2160 }(109,a) \;\) at \(\;a = \) e.g. 2