Properties

Label 1764.by
Modulus $1764$
Conductor $441$
Order $21$
Real no
Primitive no
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([0,14,12]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(85,1764))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(1764\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 441.ba
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

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

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(11\) \(13\) \(17\) \(19\) \(23\) \(25\) \(29\) \(31\) \(37\)
\(\chi_{1764}(85,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{1764}(169,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{1764}(337,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{1764}(421,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{1764}(673,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{1}{7}\right)\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{7}\right)\)
\(\chi_{1764}(841,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{1764}(925,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{4}{7}\right)\) \(1\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{4}{7}\right)\)
\(\chi_{1764}(1093,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(1\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{16}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{6}{7}\right)\)
\(\chi_{1764}(1345,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{2}{7}\right)\) \(1\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{2}{7}\right)\)
\(\chi_{1764}(1429,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{13}{21}\right)\) \(e\left(\frac{3}{7}\right)\) \(1\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{8}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{7}\right)\)
\(\chi_{1764}(1597,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{17}{21}\right)\) \(e\left(\frac{5}{7}\right)\) \(1\) \(e\left(\frac{20}{21}\right)\) \(e\left(\frac{4}{21}\right)\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{7}\right)\)
\(\chi_{1764}(1681,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{21}\right)\) \(e\left(\frac{5}{21}\right)\) \(e\left(\frac{19}{21}\right)\) \(e\left(\frac{6}{7}\right)\) \(1\) \(e\left(\frac{10}{21}\right)\) \(e\left(\frac{2}{21}\right)\) \(e\left(\frac{11}{21}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{6}{7}\right)\)