Properties

Label 1441.1083
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([52,12]))
 
pari: [g,chi] = znchar(Mod(1083,1441))
 

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.bh

\(\chi_{1441}(20,\cdot)\) \(\chi_{1441}(48,\cdot)\) \(\chi_{1441}(49,\cdot)\) \(\chi_{1441}(91,\cdot)\) \(\chi_{1441}(108,\cdot)\) \(\chi_{1441}(136,\cdot)\) \(\chi_{1441}(152,\cdot)\) \(\chi_{1441}(158,\cdot)\) \(\chi_{1441}(169,\cdot)\) \(\chi_{1441}(278,\cdot)\) \(\chi_{1441}(295,\cdot)\) \(\chi_{1441}(306,\cdot)\) \(\chi_{1441}(356,\cdot)\) \(\chi_{1441}(379,\cdot)\) \(\chi_{1441}(400,\cdot)\) \(\chi_{1441}(467,\cdot)\) \(\chi_{1441}(498,\cdot)\) \(\chi_{1441}(537,\cdot)\) \(\chi_{1441}(565,\cdot)\) \(\chi_{1441}(647,\cdot)\) \(\chi_{1441}(653,\cdot)\) \(\chi_{1441}(719,\cdot)\) \(\chi_{1441}(764,\cdot)\) \(\chi_{1441}(795,\cdot)\) \(\chi_{1441}(801,\cdot)\) \(\chi_{1441}(841,\cdot)\) \(\chi_{1441}(861,\cdot)\) \(\chi_{1441}(863,\cdot)\) \(\chi_{1441}(867,\cdot)\) \(\chi_{1441}(900,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{6}{65}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(1083, a) \) \(1\)\(1\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{64}{65}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{22}{65}\right)\)\(e\left(\frac{43}{65}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{22}{65}\right)\)\(e\left(\frac{54}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(1083,a) \;\) at \(\;a = \) e.g. 2