Properties

Label 5610.1873
Modulus $5610$
Conductor $935$
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([0,60,64,5]))
 
pari: [g,chi] = znchar(Mod(1873,5610))
 

Basic properties

Modulus: \(5610\)
Conductor: \(935\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(80\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{935}(3,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.fl

\(\chi_{5610}(367,\cdot)\) \(\chi_{5610}(487,\cdot)\) \(\chi_{5610}(643,\cdot)\) \(\chi_{5610}(907,\cdot)\) \(\chi_{5610}(1027,\cdot)\) \(\chi_{5610}(1093,\cdot)\) \(\chi_{5610}(1153,\cdot)\) \(\chi_{5610}(1417,\cdot)\) \(\chi_{5610}(1423,\cdot)\) \(\chi_{5610}(1873,\cdot)\) \(\chi_{5610}(1897,\cdot)\) \(\chi_{5610}(2017,\cdot)\) \(\chi_{5610}(2407,\cdot)\) \(\chi_{5610}(2557,\cdot)\) \(\chi_{5610}(2623,\cdot)\) \(\chi_{5610}(2953,\cdot)\) \(\chi_{5610}(3067,\cdot)\) \(\chi_{5610}(3133,\cdot)\) \(\chi_{5610}(3193,\cdot)\) \(\chi_{5610}(3403,\cdot)\) \(\chi_{5610}(3457,\cdot)\) \(\chi_{5610}(3463,\cdot)\) \(\chi_{5610}(3547,\cdot)\) \(\chi_{5610}(4057,\cdot)\) \(\chi_{5610}(4447,\cdot)\) \(\chi_{5610}(4723,\cdot)\) \(\chi_{5610}(4933,\cdot)\) \(\chi_{5610}(4987,\cdot)\) \(\chi_{5610}(5107,\cdot)\) \(\chi_{5610}(5173,\cdot)\) ...

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

\((1871,3367,1531,3301)\) → \((1,-i,e\left(\frac{4}{5}\right),e\left(\frac{1}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(1873, a) \) \(1\)\(1\)\(e\left(\frac{3}{80}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{73}{80}\right)\)\(e\left(\frac{29}{80}\right)\)\(e\left(\frac{33}{80}\right)\)\(e\left(\frac{7}{80}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{2}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(1873,a) \;\) at \(\;a = \) e.g. 2