Properties

Label 1441.47
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([104,105]))
 
pari: [g,chi] = znchar(Mod(47,1441))
 

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.bt

\(\chi_{1441}(47,\cdot)\) \(\chi_{1441}(69,\cdot)\) \(\chi_{1441}(71,\cdot)\) \(\chi_{1441}(86,\cdot)\) \(\chi_{1441}(92,\cdot)\) \(\chi_{1441}(163,\cdot)\) \(\chi_{1441}(202,\cdot)\) \(\chi_{1441}(223,\cdot)\) \(\chi_{1441}(280,\cdot)\) \(\chi_{1441}(313,\cdot)\) \(\chi_{1441}(333,\cdot)\) \(\chi_{1441}(411,\cdot)\) \(\chi_{1441}(412,\cdot)\) \(\chi_{1441}(444,\cdot)\) \(\chi_{1441}(542,\cdot)\) \(\chi_{1441}(543,\cdot)\) \(\chi_{1441}(548,\cdot)\) \(\chi_{1441}(575,\cdot)\) \(\chi_{1441}(592,\cdot)\) \(\chi_{1441}(603,\cdot)\) \(\chi_{1441}(610,\cdot)\) \(\chi_{1441}(674,\cdot)\) \(\chi_{1441}(687,\cdot)\) \(\chi_{1441}(702,\cdot)\) \(\chi_{1441}(724,\cdot)\) \(\chi_{1441}(741,\cdot)\) \(\chi_{1441}(818,\cdot)\) \(\chi_{1441}(872,\cdot)\) \(\chi_{1441}(878,\cdot)\) \(\chi_{1441}(949,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{21}{26}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(-1\)\(1\)\(e\left(\frac{79}{130}\right)\)\(e\left(\frac{36}{65}\right)\)\(e\left(\frac{14}{65}\right)\)\(e\left(\frac{23}{65}\right)\)\(e\left(\frac{21}{130}\right)\)\(e\left(\frac{9}{65}\right)\)\(e\left(\frac{107}{130}\right)\)\(e\left(\frac{7}{65}\right)\)\(e\left(\frac{25}{26}\right)\)\(e\left(\frac{10}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(47,a) \;\) at \(\;a = \) e.g. 2