Properties

Label 5610.2819
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([40,40,64,45]))
 
pari: [g,chi] = znchar(Mod(2819,5610))
 

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 \(\chi_{2805}(14,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5610.fg

\(\chi_{5610}(269,\cdot)\) \(\chi_{5610}(449,\cdot)\) \(\chi_{5610}(719,\cdot)\) \(\chi_{5610}(779,\cdot)\) \(\chi_{5610}(839,\cdot)\) \(\chi_{5610}(929,\cdot)\) \(\chi_{5610}(1049,\cdot)\) \(\chi_{5610}(1439,\cdot)\) \(\chi_{5610}(1499,\cdot)\) \(\chi_{5610}(1829,\cdot)\) \(\chi_{5610}(2249,\cdot)\) \(\chi_{5610}(2369,\cdot)\) \(\chi_{5610}(2489,\cdot)\) \(\chi_{5610}(2579,\cdot)\) \(\chi_{5610}(2759,\cdot)\) \(\chi_{5610}(2819,\cdot)\) \(\chi_{5610}(3029,\cdot)\) \(\chi_{5610}(3089,\cdot)\) \(\chi_{5610}(3359,\cdot)\) \(\chi_{5610}(3479,\cdot)\) \(\chi_{5610}(3899,\cdot)\) \(\chi_{5610}(4019,\cdot)\) \(\chi_{5610}(4349,\cdot)\) \(\chi_{5610}(4409,\cdot)\) \(\chi_{5610}(4559,\cdot)\) \(\chi_{5610}(4799,\cdot)\) \(\chi_{5610}(4889,\cdot)\) \(\chi_{5610}(5009,\cdot)\) \(\chi_{5610}(5069,\cdot)\) \(\chi_{5610}(5129,\cdot)\) ...

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

\((1871,3367,1531,3301)\) → \((-1,-1,e\left(\frac{4}{5}\right),e\left(\frac{9}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(13\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 5610 }(2819, a) \) \(1\)\(1\)\(e\left(\frac{23}{80}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{33}{80}\right)\)\(e\left(\frac{69}{80}\right)\)\(e\left(\frac{53}{80}\right)\)\(e\left(\frac{7}{80}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{13}{20}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5610 }(2819,a) \;\) at \(\;a = \) e.g. 2