Properties

Label 1764.643
Modulus $1764$
Conductor $1764$
Order $42$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,14,27]))
 
pari: [g,chi] = znchar(Mod(643,1764))
 

Basic properties

Modulus: \(1764\)
Conductor: \(1764\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 1764.cf

\(\chi_{1764}(139,\cdot)\) \(\chi_{1764}(223,\cdot)\) \(\chi_{1764}(475,\cdot)\) \(\chi_{1764}(643,\cdot)\) \(\chi_{1764}(727,\cdot)\) \(\chi_{1764}(895,\cdot)\) \(\chi_{1764}(1147,\cdot)\) \(\chi_{1764}(1231,\cdot)\) \(\chi_{1764}(1399,\cdot)\) \(\chi_{1764}(1483,\cdot)\) \(\chi_{1764}(1651,\cdot)\) \(\chi_{1764}(1735,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((883,785,1081)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{9}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1764 }(643, a) \) \(1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{1}{14}\right)\)\(1\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1764 }(643,a) \;\) at \(\;a = \) e.g. 2