Properties

Label 1441.1061
Modulus $1441$
Conductor $1441$
Order $65$
Real no
Primitive yes
Minimal yes
Parity even

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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([52,18]))
 
pari: [g,chi] = znchar(Mod(1061,1441))
 

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.bg

\(\chi_{1441}(36,\cdot)\) \(\chi_{1441}(102,\cdot)\) \(\chi_{1441}(147,\cdot)\) \(\chi_{1441}(159,\cdot)\) \(\chi_{1441}(179,\cdot)\) \(\chi_{1441}(240,\cdot)\) \(\chi_{1441}(267,\cdot)\) \(\chi_{1441}(289,\cdot)\) \(\chi_{1441}(300,\cdot)\) \(\chi_{1441}(317,\cdot)\) \(\chi_{1441}(339,\cdot)\) \(\chi_{1441}(367,\cdot)\) \(\chi_{1441}(383,\cdot)\) \(\chi_{1441}(487,\cdot)\) \(\chi_{1441}(531,\cdot)\) \(\chi_{1441}(544,\cdot)\) \(\chi_{1441}(559,\cdot)\) \(\chi_{1441}(632,\cdot)\) \(\chi_{1441}(658,\cdot)\) \(\chi_{1441}(664,\cdot)\) \(\chi_{1441}(676,\cdot)\) \(\chi_{1441}(680,\cdot)\) \(\chi_{1441}(696,\cdot)\) \(\chi_{1441}(730,\cdot)\) \(\chi_{1441}(784,\cdot)\) \(\chi_{1441}(790,\cdot)\) \(\chi_{1441}(819,\cdot)\) \(\chi_{1441}(829,\cdot)\) \(\chi_{1441}(830,\cdot)\) \(\chi_{1441}(845,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{9}{65}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(1061, a) \) \(1\)\(1\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{11}{65}\right)\)\(e\left(\frac{1}{13}\right)\)\(e\left(\frac{63}{65}\right)\)\(e\left(\frac{46}{65}\right)\)\(e\left(\frac{6}{65}\right)\)\(e\left(\frac{8}{13}\right)\)\(e\left(\frac{22}{65}\right)\)\(e\left(\frac{33}{65}\right)\)\(e\left(\frac{16}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(1061,a) \;\) at \(\;a = \) e.g. 2