Properties

Label 1441.494
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
Minimal yes
Parity odd

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([65,54]))
 
pari: [g,chi] = znchar(Mod(494,1441))
 

Basic properties

Modulus: \(1441\)
Conductor: \(1441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(130\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.bq

\(\chi_{1441}(21,\cdot)\) \(\chi_{1441}(43,\cdot)\) \(\chi_{1441}(65,\cdot)\) \(\chi_{1441}(109,\cdot)\) \(\chi_{1441}(142,\cdot)\) \(\chi_{1441}(164,\cdot)\) \(\chi_{1441}(175,\cdot)\) \(\chi_{1441}(186,\cdot)\) \(\chi_{1441}(208,\cdot)\) \(\chi_{1441}(252,\cdot)\) \(\chi_{1441}(274,\cdot)\) \(\chi_{1441}(296,\cdot)\) \(\chi_{1441}(362,\cdot)\) \(\chi_{1441}(406,\cdot)\) \(\chi_{1441}(428,\cdot)\) \(\chi_{1441}(439,\cdot)\) \(\chi_{1441}(494,\cdot)\) \(\chi_{1441}(516,\cdot)\) \(\chi_{1441}(527,\cdot)\) \(\chi_{1441}(549,\cdot)\) \(\chi_{1441}(560,\cdot)\) \(\chi_{1441}(615,\cdot)\) \(\chi_{1441}(626,\cdot)\) \(\chi_{1441}(659,\cdot)\) \(\chi_{1441}(670,\cdot)\) \(\chi_{1441}(703,\cdot)\) \(\chi_{1441}(714,\cdot)\) \(\chi_{1441}(736,\cdot)\) \(\chi_{1441}(769,\cdot)\) \(\chi_{1441}(780,\cdot)\) ...

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 130 polynomial (not computed)

Values on generators

\((1311,133)\) → \((-1,e\left(\frac{27}{65}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{119}{130}\right)\)\(e\left(\frac{59}{65}\right)\)\(e\left(\frac{54}{65}\right)\)\(e\left(\frac{7}{65}\right)\)\(e\left(\frac{107}{130}\right)\)\(e\left(\frac{49}{130}\right)\)\(e\left(\frac{97}{130}\right)\)\(e\left(\frac{53}{65}\right)\)\(e\left(\frac{3}{130}\right)\)\(e\left(\frac{48}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(494,a) \;\) at \(\;a = \) e.g. 2