Properties

Label 1764.83
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([21,7,9]))
 
pari: [g,chi] = znchar(Mod(83,1764))
 

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

Values on generators

\((883,785,1081)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{3}{14}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(1\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{2}{21}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{6}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.12140927516266774666314598865493926760663766983493911125885658849232780316031818617435329218169600416415744.1