Properties

Label 1764.143
Modulus $1764$
Conductor $588$
Order $42$
Real no
Primitive no
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([21,21,31]))
 
pari: [g,chi] = znchar(Mod(143,1764))
 

Basic properties

Modulus: \(1764\)
Conductor: \(588\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{588}(143,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1764.cx

\(\chi_{1764}(143,\cdot)\) \(\chi_{1764}(395,\cdot)\) \(\chi_{1764}(467,\cdot)\) \(\chi_{1764}(647,\cdot)\) \(\chi_{1764}(719,\cdot)\) \(\chi_{1764}(899,\cdot)\) \(\chi_{1764}(971,\cdot)\) \(\chi_{1764}(1151,\cdot)\) \(\chi_{1764}(1223,\cdot)\) \(\chi_{1764}(1475,\cdot)\) \(\chi_{1764}(1655,\cdot)\) \(\chi_{1764}(1727,\cdot)\)

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

\((883,785,1081)\) → \((-1,-1,e\left(\frac{31}{42}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{13}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.5436948860695888782893198886016377149049148530413040928765325951335574011833955525853184.1