Properties

Label 1441.434
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([52,126]))
 
pari: [g,chi] = znchar(Mod(434,1441))
 

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.bj

\(\chi_{1441}(3,\cdot)\) \(\chi_{1441}(5,\cdot)\) \(\chi_{1441}(9,\cdot)\) \(\chi_{1441}(15,\cdot)\) \(\chi_{1441}(25,\cdot)\) \(\chi_{1441}(27,\cdot)\) \(\chi_{1441}(38,\cdot)\) \(\chi_{1441}(75,\cdot)\) \(\chi_{1441}(81,\cdot)\) \(\chi_{1441}(114,\cdot)\) \(\chi_{1441}(125,\cdot)\) \(\chi_{1441}(135,\cdot)\) \(\chi_{1441}(174,\cdot)\) \(\chi_{1441}(190,\cdot)\) \(\chi_{1441}(196,\cdot)\) \(\chi_{1441}(225,\cdot)\) \(\chi_{1441}(269,\cdot)\) \(\chi_{1441}(290,\cdot)\) \(\chi_{1441}(311,\cdot)\) \(\chi_{1441}(405,\cdot)\) \(\chi_{1441}(427,\cdot)\) \(\chi_{1441}(434,\cdot)\) \(\chi_{1441}(493,\cdot)\) \(\chi_{1441}(522,\cdot)\) \(\chi_{1441}(570,\cdot)\) \(\chi_{1441}(588,\cdot)\) \(\chi_{1441}(625,\cdot)\) \(\chi_{1441}(641,\cdot)\) \(\chi_{1441}(675,\cdot)\) \(\chi_{1441}(729,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{63}{65}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{24}{65}\right)\)\(e\left(\frac{64}{65}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{12}{65}\right)\)\(e\left(\frac{23}{65}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{7}{65}\right)\)\(e\left(\frac{63}{65}\right)\)\(e\left(\frac{36}{65}\right)\)\(e\left(\frac{47}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(434,a) \;\) at \(\;a = \) e.g. 2