Properties

Label 1764.1679
Modulus $1764$
Conductor $1764$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

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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([21,35,33]))
 
pari: [g,chi] = znchar(Mod(1679,1764))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1764.cv

\(\chi_{1764}(83,\cdot)\) \(\chi_{1764}(167,\cdot)\) \(\chi_{1764}(335,\cdot)\) \(\chi_{1764}(419,\cdot)\) \(\chi_{1764}(671,\cdot)\) \(\chi_{1764}(839,\cdot)\) \(\chi_{1764}(923,\cdot)\) \(\chi_{1764}(1091,\cdot)\) \(\chi_{1764}(1343,\cdot)\) \(\chi_{1764}(1427,\cdot)\) \(\chi_{1764}(1595,\cdot)\) \(\chi_{1764}(1679,\cdot)\)

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: 42.0.12140927516266774666314598865493926760663766983493911125885658849232780316031818617435329218169600416415744.1

Values on generators

\((883,785,1081)\) → \((-1,e\left(\frac{5}{6}\right),e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1764 }(1679, a) \) \(-1\)\(1\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{1}{7}\right)\)\(1\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1764 }(1679,a) \;\) at \(\;a = \) e.g. 2