Properties

Label 1568.113
Modulus $1568$
Conductor $392$
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(1568, base_ring=CyclotomicField(14))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7,10]))
 
pari: [g,chi] = znchar(Mod(113,1568))
 

Basic properties

Modulus: \(1568\)
Conductor: \(392\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{392}(309,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1568.bj

\(\chi_{1568}(113,\cdot)\) \(\chi_{1568}(337,\cdot)\) \(\chi_{1568}(561,\cdot)\) \(\chi_{1568}(1009,\cdot)\) \(\chi_{1568}(1233,\cdot)\) \(\chi_{1568}(1457,\cdot)\)

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

\((1471,197,1473)\) → \((1,-1,e\left(\frac{5}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)
\( \chi_{ 1568 }(113, a) \) \(1\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(-1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1568 }(113,a) \;\) at \(\;a = \) e.g. 2