Properties

Label 1176.71
Modulus $1176$
Conductor $588$
Order $14$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1176, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,0,7,8]))
 
pari: [g,chi] = znchar(Mod(71,1176))
 

Basic properties

Modulus: \(1176\)
Conductor: \(588\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{588}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1176.bp

\(\chi_{1176}(71,\cdot)\) \(\chi_{1176}(239,\cdot)\) \(\chi_{1176}(407,\cdot)\) \(\chi_{1176}(575,\cdot)\) \(\chi_{1176}(743,\cdot)\) \(\chi_{1176}(911,\cdot)\)

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: \(\Q(\zeta_{7})\)
Fixed field: Number field defined by a degree 14 polynomial

Values on generators

\((295,589,785,1081)\) → \((-1,1,-1,e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1176 }(71, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(-1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(-1\)\(e\left(\frac{2}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1176 }(71,a) \;\) at \(\;a = \) e.g. 2