Properties

Label 5600.19
Modulus $5600$
Conductor $5600$
Order $120$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5600, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([60,105,108,100]))
 
pari: [g,chi] = znchar(Mod(19,5600))
 

Basic properties

Modulus: \(5600\)
Conductor: \(5600\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5600.id

\(\chi_{5600}(19,\cdot)\) \(\chi_{5600}(59,\cdot)\) \(\chi_{5600}(339,\cdot)\) \(\chi_{5600}(579,\cdot)\) \(\chi_{5600}(619,\cdot)\) \(\chi_{5600}(859,\cdot)\) \(\chi_{5600}(1139,\cdot)\) \(\chi_{5600}(1179,\cdot)\) \(\chi_{5600}(1419,\cdot)\) \(\chi_{5600}(1459,\cdot)\) \(\chi_{5600}(1739,\cdot)\) \(\chi_{5600}(1979,\cdot)\) \(\chi_{5600}(2019,\cdot)\) \(\chi_{5600}(2259,\cdot)\) \(\chi_{5600}(2539,\cdot)\) \(\chi_{5600}(2579,\cdot)\) \(\chi_{5600}(2819,\cdot)\) \(\chi_{5600}(2859,\cdot)\) \(\chi_{5600}(3139,\cdot)\) \(\chi_{5600}(3379,\cdot)\) \(\chi_{5600}(3419,\cdot)\) \(\chi_{5600}(3659,\cdot)\) \(\chi_{5600}(3939,\cdot)\) \(\chi_{5600}(3979,\cdot)\) \(\chi_{5600}(4219,\cdot)\) \(\chi_{5600}(4259,\cdot)\) \(\chi_{5600}(4539,\cdot)\) \(\chi_{5600}(4779,\cdot)\) \(\chi_{5600}(4819,\cdot)\) \(\chi_{5600}(5059,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((351,4901,5377,801)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{9}{10}\right),e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 5600 }(19, a) \) \(1\)\(1\)\(e\left(\frac{31}{120}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{73}{120}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{119}{120}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{8}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5600 }(19,a) \;\) at \(\;a = \) e.g. 2