Properties

Label 1764.163
Modulus $1764$
Conductor $196$
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([21,0,20]))
 
pari: [g,chi] = znchar(Mod(163,1764))
 

Basic properties

Modulus: \(1764\)
Conductor: \(196\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{196}(163,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1764.cp

\(\chi_{1764}(163,\cdot)\) \(\chi_{1764}(235,\cdot)\) \(\chi_{1764}(415,\cdot)\) \(\chi_{1764}(487,\cdot)\) \(\chi_{1764}(739,\cdot)\) \(\chi_{1764}(919,\cdot)\) \(\chi_{1764}(991,\cdot)\) \(\chi_{1764}(1171,\cdot)\) \(\chi_{1764}(1423,\cdot)\) \(\chi_{1764}(1495,\cdot)\) \(\chi_{1764}(1675,\cdot)\) \(\chi_{1764}(1747,\cdot)\)

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

\((883,785,1081)\) → \((-1,1,e\left(\frac{10}{21}\right))\)

Values

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

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.0.74252462132603256348231837398371002884673933378885582779211491265789772693504.1