Properties

Label 4719.874
Modulus $4719$
Conductor $143$
Order $15$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4719, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,12,10]))
 
pari: [g,chi] = znchar(Mod(874,4719))
 

Basic properties

Modulus: \(4719\)
Conductor: \(143\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{143}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4719.bh

\(\chi_{4719}(874,\cdot)\) \(\chi_{4719}(1576,\cdot)\) \(\chi_{4719}(1582,\cdot)\) \(\chi_{4719}(1654,\cdot)\) \(\chi_{4719}(3415,\cdot)\) \(\chi_{4719}(3760,\cdot)\) \(\chi_{4719}(4117,\cdot)\) \(\chi_{4719}(4195,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\((1574,3511,4357)\) → \((1,e\left(\frac{2}{5}\right),e\left(\frac{1}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4719 }(874, a) \) \(1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{13}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4719 }(874,a) \;\) at \(\;a = \) e.g. 2