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Properties

Label 1110.697
Modulus $1110$
Conductor $185$
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(1110, base_ring=CyclotomicField(4)) M = H._module chi = DirichletCharacter(H, M([0,1,1]))
 
Copy content pari:[g,chi] = znchar(Mod(697,1110))
 

Basic properties

Modulus: \(1110\)
Conductor: \(185\)
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 \(\chi_{185}(142,\cdot)\)
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 1110.l

\(\chi_{1110}(43,\cdot)\) \(\chi_{1110}(697,\cdot)\)

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: 4.4.6331625.1

Values on generators

\((371,667,631)\) → \((1,i,i)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 1110 }(697, a) \) \(1\)\(1\)\(i\)\(-1\)\(-1\)\(1\)\(i\)\(-1\)\(-i\)\(i\)\(-1\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1110 }(697,a) \;\) at \(\;a = \) e.g. 2