Properties

Label 1764.1019
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([21,7,34]))
 
pari: [g,chi] = znchar(Mod(1019,1764))
 

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.cm

\(\chi_{1764}(11,\cdot)\) \(\chi_{1764}(23,\cdot)\) \(\chi_{1764}(515,\cdot)\) \(\chi_{1764}(527,\cdot)\) \(\chi_{1764}(767,\cdot)\) \(\chi_{1764}(779,\cdot)\) \(\chi_{1764}(1019,\cdot)\) \(\chi_{1764}(1031,\cdot)\) \(\chi_{1764}(1271,\cdot)\) \(\chi_{1764}(1283,\cdot)\) \(\chi_{1764}(1523,\cdot)\) \(\chi_{1764}(1535,\cdot)\)

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: 42.42.84986492613867422664202192058457487324646368884457377881199611944629462212222730322047304527187202914910208.1

Values on generators

\((883,785,1081)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1764 }(1019, a) \) \(1\)\(1\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{31}{42}\right)\)\(-1\)\(e\left(\frac{19}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1764 }(1019,a) \;\) at \(\;a = \) e.g. 2