Properties

Label 1764.181
Modulus $1764$
Conductor $49$
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([0,0,3]))
 
pari: [g,chi] = znchar(Mod(181,1764))
 

Basic properties

Modulus: \(1764\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(14\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1764.bt

\(\chi_{1764}(181,\cdot)\) \(\chi_{1764}(433,\cdot)\) \(\chi_{1764}(937,\cdot)\) \(\chi_{1764}(1189,\cdot)\) \(\chi_{1764}(1441,\cdot)\) \(\chi_{1764}(1693,\cdot)\)

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

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

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(-1\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(-1\)\(e\left(\frac{6}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{7})\)
Fixed field: 14.0.1341068619663964900807.1