Properties

Label 1764.251
Modulus $1764$
Conductor $588$
Order $14$
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([7,7,9]))
 
pari: [g,chi] = znchar(Mod(251,1764))
 

Basic properties

Modulus: \(1764\)
Conductor: \(588\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{588}(251,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1764.bp

\(\chi_{1764}(251,\cdot)\) \(\chi_{1764}(503,\cdot)\) \(\chi_{1764}(755,\cdot)\) \(\chi_{1764}(1007,\cdot)\) \(\chi_{1764}(1259,\cdot)\) \(\chi_{1764}(1511,\cdot)\)

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

\((883,785,1081)\) → \((-1,-1,e\left(\frac{9}{14}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(1\)\(e\left(\frac{4}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 14.0.48052913294624214844455469056.1