Properties

Label 4851.37
Modulus $4851$
Conductor $539$
Order $105$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4851, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,160,42]))
 
pari: [g,chi] = znchar(Mod(37,4851))
 

Basic properties

Modulus: \(4851\)
Conductor: \(539\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{539}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4851.ez

\(\chi_{4851}(37,\cdot)\) \(\chi_{4851}(163,\cdot)\) \(\chi_{4851}(235,\cdot)\) \(\chi_{4851}(289,\cdot)\) \(\chi_{4851}(478,\cdot)\) \(\chi_{4851}(487,\cdot)\) \(\chi_{4851}(676,\cdot)\) \(\chi_{4851}(730,\cdot)\) \(\chi_{4851}(856,\cdot)\) \(\chi_{4851}(928,\cdot)\) \(\chi_{4851}(982,\cdot)\) \(\chi_{4851}(1054,\cdot)\) \(\chi_{4851}(1171,\cdot)\) \(\chi_{4851}(1180,\cdot)\) \(\chi_{4851}(1369,\cdot)\) \(\chi_{4851}(1423,\cdot)\) \(\chi_{4851}(1621,\cdot)\) \(\chi_{4851}(1675,\cdot)\) \(\chi_{4851}(1747,\cdot)\) \(\chi_{4851}(1864,\cdot)\) \(\chi_{4851}(1873,\cdot)\) \(\chi_{4851}(2062,\cdot)\) \(\chi_{4851}(2116,\cdot)\) \(\chi_{4851}(2242,\cdot)\) \(\chi_{4851}(2314,\cdot)\) \(\chi_{4851}(2368,\cdot)\) \(\chi_{4851}(2440,\cdot)\) \(\chi_{4851}(2557,\cdot)\) \(\chi_{4851}(2755,\cdot)\) \(\chi_{4851}(2809,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 105 polynomial (not computed)

Values on generators

\((4313,199,442)\) → \((1,e\left(\frac{16}{21}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 4851 }(37, a) \) \(1\)\(1\)\(e\left(\frac{1}{105}\right)\)\(e\left(\frac{2}{105}\right)\)\(e\left(\frac{94}{105}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{4}{105}\right)\)\(e\left(\frac{89}{105}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{32}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4851 }(37,a) \;\) at \(\;a = \) e.g. 2