Properties

Label 8470.5727
Modulus $8470$
Conductor $55$
Order $20$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8470, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,0,14]))
 
pari: [g,chi] = znchar(Mod(5727,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(55\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{55}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.bi

\(\chi_{8470}(1443,\cdot)\) \(\chi_{8470}(1667,\cdot)\) \(\chi_{8470}(1933,\cdot)\) \(\chi_{8470}(3137,\cdot)\) \(\chi_{8470}(3627,\cdot)\) \(\chi_{8470}(4033,\cdot)\) \(\chi_{8470}(5727,\cdot)\) \(\chi_{8470}(8443,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: \(\Q(\zeta_{55})^+\)

Values on generators

\((6777,6051,7141)\) → \((i,1,e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\( \chi_{ 8470 }(5727, a) \) \(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(-i\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{13}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(5727,a) \;\) at \(\;a = \) e.g. 2