Properties

Label 5610.53
Modulus $5610$
Conductor $2805$
Order $40$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5610, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([20,30,24,35]))
 
pari: [g,chi] = znchar(Mod(53,5610))
 

Basic properties

Modulus: \(5610\)
Conductor: \(2805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2805}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.eq

\(\chi_{5610}(53,\cdot)\) \(\chi_{5610}(467,\cdot)\) \(\chi_{5610}(587,\cdot)\) \(\chi_{5610}(977,\cdot)\) \(\chi_{5610}(1103,\cdot)\) \(\chi_{5610}(2093,\cdot)\) \(\chi_{5610}(2117,\cdot)\) \(\chi_{5610}(2627,\cdot)\) \(\chi_{5610}(2633,\cdot)\) \(\chi_{5610}(3017,\cdot)\) \(\chi_{5610}(3623,\cdot)\) \(\chi_{5610}(4163,\cdot)\) \(\chi_{5610}(4547,\cdot)\) \(\chi_{5610}(4667,\cdot)\) \(\chi_{5610}(4673,\cdot)\) \(\chi_{5610}(5153,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((1871,3367,1531,3301)\) → \((-1,-i,e\left(\frac{3}{5}\right),e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(53, a) \) \(1\)\(1\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{37}{40}\right)\)\(1\)\(e\left(\frac{11}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(53,a) \;\) at \(\;a = \) e.g. 2