Properties

Label 1050.101
Modulus $1050$
Conductor $21$
Order $6$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([3,0,1]))
 
pari: [g,chi] = znchar(Mod(101,1050))
 

Basic properties

Modulus: \(1050\)
Conductor: \(21\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{21}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1050.s

\(\chi_{1050}(101,\cdot)\) \(\chi_{1050}(551,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: \(\Q(\zeta_{21})^+\)

Values on generators

\((701,127,451)\) → \((-1,1,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1050 }(101, a) \) \(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{6}\right)\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1050 }(101,a) \;\) at \(\;a = \) e.g. 2