Properties

Label 1764.1075
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([21,14,22]))
 
pari: [g,chi] = znchar(Mod(1075,1764))
 

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.cc

\(\chi_{1764}(319,\cdot)\) \(\chi_{1764}(331,\cdot)\) \(\chi_{1764}(571,\cdot)\) \(\chi_{1764}(583,\cdot)\) \(\chi_{1764}(823,\cdot)\) \(\chi_{1764}(835,\cdot)\) \(\chi_{1764}(1075,\cdot)\) \(\chi_{1764}(1087,\cdot)\) \(\chi_{1764}(1327,\cdot)\) \(\chi_{1764}(1339,\cdot)\) \(\chi_{1764}(1579,\cdot)\) \(\chi_{1764}(1591,\cdot)\)

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: 42.0.38859850303551633591313302267241649439710273838343565560676548671526960316517023466871195485682305859584.2

Values on generators

\((883,785,1081)\) → \((-1,e\left(\frac{1}{3}\right),e\left(\frac{11}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1764 }(1075, a) \) \(-1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{16}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1764 }(1075,a) \;\) at \(\;a = \) e.g. 2