Properties

Label 1764.43
Modulus $1764$
Conductor $1764$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1764)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,28,6]))
 
pari: [g,chi] = znchar(Mod(43,1764))
 

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.cs

\(\chi_{1764}(43,\cdot)\) \(\chi_{1764}(211,\cdot)\) \(\chi_{1764}(463,\cdot)\) \(\chi_{1764}(547,\cdot)\) \(\chi_{1764}(715,\cdot)\) \(\chi_{1764}(799,\cdot)\) \(\chi_{1764}(967,\cdot)\) \(\chi_{1764}(1051,\cdot)\) \(\chi_{1764}(1219,\cdot)\) \(\chi_{1764}(1303,\cdot)\) \(\chi_{1764}(1555,\cdot)\) \(\chi_{1764}(1723,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((883,785,1081)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{1}{7}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(-1\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{4}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.16184860601229335106752728974278071403461171944333013561298021104342757316333620769209160968630697984.1