Properties

Label 1716.703
Modulus $1716$
Conductor $44$
Order $2$
Real yes
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1716, base_ring=CyclotomicField(2))
 
M = H._module
 
chi = DirichletCharacter(H, M([1,0,1,0]))
 
pari: [g,chi] = znchar(Mod(703,1716))
 

Basic properties

Modulus: \(1716\)
Conductor: \(44\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: yes
Primitive: no, induced from \(\chi_{44}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1716.n

\(\chi_{1716}(703,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q\)
Fixed field: \(\Q(\sqrt{11}) \)

Values on generators

\((859,1145,937,925)\) → \((-1,1,-1,1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)\(37\)
\( \chi_{ 1716 }(703, a) \) \(1\)\(1\)\(1\)\(1\)\(-1\)\(1\)\(-1\)\(1\)\(-1\)\(-1\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1716 }(703,a) \;\) at \(\;a = \) e.g. 2