Properties

Label 4928.51
Modulus $4928$
Conductor $4928$
Order $240$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4928, base_ring=CyclotomicField(240))
 
M = H._module
 
chi = DirichletCharacter(H, M([120,225,80,168]))
 
pari: [g,chi] = znchar(Mod(51,4928))
 

Basic properties

Modulus: \(4928\)
Conductor: \(4928\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(240\)
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 4928.gc

\(\chi_{4928}(51,\cdot)\) \(\chi_{4928}(107,\cdot)\) \(\chi_{4928}(123,\cdot)\) \(\chi_{4928}(347,\cdot)\) \(\chi_{4928}(387,\cdot)\) \(\chi_{4928}(403,\cdot)\) \(\chi_{4928}(459,\cdot)\) \(\chi_{4928}(611,\cdot)\) \(\chi_{4928}(667,\cdot)\) \(\chi_{4928}(723,\cdot)\) \(\chi_{4928}(739,\cdot)\) \(\chi_{4928}(963,\cdot)\) \(\chi_{4928}(1003,\cdot)\) \(\chi_{4928}(1019,\cdot)\) \(\chi_{4928}(1075,\cdot)\) \(\chi_{4928}(1227,\cdot)\) \(\chi_{4928}(1283,\cdot)\) \(\chi_{4928}(1339,\cdot)\) \(\chi_{4928}(1355,\cdot)\) \(\chi_{4928}(1579,\cdot)\) \(\chi_{4928}(1619,\cdot)\) \(\chi_{4928}(1635,\cdot)\) \(\chi_{4928}(1691,\cdot)\) \(\chi_{4928}(1843,\cdot)\) \(\chi_{4928}(1899,\cdot)\) \(\chi_{4928}(1955,\cdot)\) \(\chi_{4928}(1971,\cdot)\) \(\chi_{4928}(2195,\cdot)\) \(\chi_{4928}(2235,\cdot)\) \(\chi_{4928}(2251,\cdot)\) ...

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: $\Q(\zeta_{240})$
Fixed field: Number field defined by a degree 240 polynomial (not computed)

Values on generators

\((4159,1541,2817,3137)\) → \((-1,e\left(\frac{15}{16}\right),e\left(\frac{1}{3}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 4928 }(51, a) \) \(1\)\(1\)\(e\left(\frac{59}{240}\right)\)\(e\left(\frac{97}{240}\right)\)\(e\left(\frac{59}{120}\right)\)\(e\left(\frac{61}{80}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{199}{240}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{97}{120}\right)\)\(e\left(\frac{59}{80}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4928 }(51,a) \;\) at \(\;a = \) e.g. 2