Properties

Label 1441.16
Modulus $1441$
Conductor $1441$
Order $65$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1441, base_ring=CyclotomicField(130))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([52,4]))
 
pari: [g,chi] = znchar(Mod(16,1441))
 

Basic properties

Modulus: \(1441\)
Conductor: \(1441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(65\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.bi

\(\chi_{1441}(4,\cdot)\) \(\chi_{1441}(16,\cdot)\) \(\chi_{1441}(59,\cdot)\) \(\chi_{1441}(64,\cdot)\) \(\chi_{1441}(146,\cdot)\) \(\chi_{1441}(180,\cdot)\) \(\chi_{1441}(212,\cdot)\) \(\chi_{1441}(236,\cdot)\) \(\chi_{1441}(245,\cdot)\) \(\chi_{1441}(256,\cdot)\) \(\chi_{1441}(273,\cdot)\) \(\chi_{1441}(421,\cdot)\) \(\chi_{1441}(510,\cdot)\) \(\chi_{1441}(533,\cdot)\) \(\chi_{1441}(598,\cdot)\) \(\chi_{1441}(599,\cdot)\) \(\chi_{1441}(691,\cdot)\) \(\chi_{1441}(698,\cdot)\) \(\chi_{1441}(720,\cdot)\) \(\chi_{1441}(757,\cdot)\) \(\chi_{1441}(806,\cdot)\) \(\chi_{1441}(834,\cdot)\) \(\chi_{1441}(894,\cdot)\) \(\chi_{1441}(895,\cdot)\) \(\chi_{1441}(922,\cdot)\) \(\chi_{1441}(929,\cdot)\) \(\chi_{1441}(938,\cdot)\) \(\chi_{1441}(944,\cdot)\) \(\chi_{1441}(951,\cdot)\) \(\chi_{1441}(955,\cdot)\) ...

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

Related number fields

Field of values: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 65 polynomial

Values on generators

\((1311,133)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{2}{65}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{28}{65}\right)\)\(e\left(\frac{27}{65}\right)\)\(e\left(\frac{56}{65}\right)\)\(e\left(\frac{1}{65}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{49}{65}\right)\)\(e\left(\frac{19}{65}\right)\)\(e\left(\frac{54}{65}\right)\)\(e\left(\frac{29}{65}\right)\)\(e\left(\frac{18}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(16,a) \;\) at \(\;a = \) e.g. 2