Properties

Label 5610.1711
Modulus $5610$
Conductor $187$
Order $80$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,72,35]))
 
pari: [g,chi] = znchar(Mod(1711,5610))
 

Basic properties

Modulus: \(5610\)
Conductor: \(187\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{187}(28,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.fc

\(\chi_{5610}(61,\cdot)\) \(\chi_{5610}(211,\cdot)\) \(\chi_{5610}(481,\cdot)\) \(\chi_{5610}(541,\cdot)\) \(\chi_{5610}(601,\cdot)\) \(\chi_{5610}(721,\cdot)\) \(\chi_{5610}(811,\cdot)\) \(\chi_{5610}(1051,\cdot)\) \(\chi_{5610}(1201,\cdot)\) \(\chi_{5610}(1261,\cdot)\) \(\chi_{5610}(1591,\cdot)\) \(\chi_{5610}(1711,\cdot)\) \(\chi_{5610}(2131,\cdot)\) \(\chi_{5610}(2251,\cdot)\) \(\chi_{5610}(2521,\cdot)\) \(\chi_{5610}(2581,\cdot)\) \(\chi_{5610}(2791,\cdot)\) \(\chi_{5610}(2851,\cdot)\) \(\chi_{5610}(3031,\cdot)\) \(\chi_{5610}(3121,\cdot)\) \(\chi_{5610}(3241,\cdot)\) \(\chi_{5610}(3361,\cdot)\) \(\chi_{5610}(3781,\cdot)\) \(\chi_{5610}(4111,\cdot)\) \(\chi_{5610}(4171,\cdot)\) \(\chi_{5610}(4561,\cdot)\) \(\chi_{5610}(4681,\cdot)\) \(\chi_{5610}(4771,\cdot)\) \(\chi_{5610}(4831,\cdot)\) \(\chi_{5610}(4891,\cdot)\) ...

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: $\Q(\zeta_{80})$
Fixed field: Number field defined by a degree 80 polynomial

Values on generators

\((1871,3367,1531,3301)\) → \((1,1,e\left(\frac{9}{10}\right),e\left(\frac{7}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(1711, a) \) \(1\)\(1\)\(e\left(\frac{9}{80}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{79}{80}\right)\)\(e\left(\frac{27}{80}\right)\)\(e\left(\frac{19}{80}\right)\)\(e\left(\frac{41}{80}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{19}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(1711,a) \;\) at \(\;a = \) e.g. 2