Properties

Label 5610.1943
Modulus $5610$
Conductor $2805$
Order $80$
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(80))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,60,56,25]))
 
pari: [g,chi] = znchar(Mod(1943,5610))
 

Basic properties

Modulus: \(5610\)
Conductor: \(2805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2805}(1943,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.fb

\(\chi_{5610}(173,\cdot)\) \(\chi_{5610}(227,\cdot)\) \(\chi_{5610}(233,\cdot)\) \(\chi_{5610}(347,\cdot)\) \(\chi_{5610}(413,\cdot)\) \(\chi_{5610}(743,\cdot)\) \(\chi_{5610}(827,\cdot)\) \(\chi_{5610}(1217,\cdot)\) \(\chi_{5610}(1337,\cdot)\) \(\chi_{5610}(1493,\cdot)\) \(\chi_{5610}(1757,\cdot)\) \(\chi_{5610}(1877,\cdot)\) \(\chi_{5610}(1943,\cdot)\) \(\chi_{5610}(2213,\cdot)\) \(\chi_{5610}(2273,\cdot)\) \(\chi_{5610}(2723,\cdot)\) \(\chi_{5610}(2747,\cdot)\) \(\chi_{5610}(2867,\cdot)\) \(\chi_{5610}(3407,\cdot)\) \(\chi_{5610}(3473,\cdot)\) \(\chi_{5610}(3533,\cdot)\) \(\chi_{5610}(3797,\cdot)\) \(\chi_{5610}(3803,\cdot)\) \(\chi_{5610}(4043,\cdot)\) \(\chi_{5610}(4253,\cdot)\) \(\chi_{5610}(4307,\cdot)\) \(\chi_{5610}(4397,\cdot)\) \(\chi_{5610}(4787,\cdot)\) \(\chi_{5610}(5297,\cdot)\) \(\chi_{5610}(5447,\cdot)\) ...

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

\((1871,3367,1531,3301)\) → \((-1,-i,e\left(\frac{7}{10}\right),e\left(\frac{5}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(1943, a) \) \(1\)\(1\)\(e\left(\frac{7}{80}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{39}{40}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{77}{80}\right)\)\(e\left(\frac{1}{80}\right)\)\(e\left(\frac{37}{80}\right)\)\(e\left(\frac{3}{80}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(1943,a) \;\) at \(\;a = \) e.g. 2