Properties

Label 1441.17
Modulus $1441$
Conductor $1441$
Order $130$
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([117,43]))
 
pari: [g,chi] = znchar(Mod(17,1441))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.bm

\(\chi_{1441}(6,\cdot)\) \(\chi_{1441}(17,\cdot)\) \(\chi_{1441}(40,\cdot)\) \(\chi_{1441}(50,\cdot)\) \(\chi_{1441}(57,\cdot)\) \(\chi_{1441}(116,\cdot)\) \(\chi_{1441}(145,\cdot)\) \(\chi_{1441}(161,\cdot)\) \(\chi_{1441}(216,\cdot)\) \(\chi_{1441}(270,\cdot)\) \(\chi_{1441}(293,\cdot)\) \(\chi_{1441}(359,\cdot)\) \(\chi_{1441}(380,\cdot)\) \(\chi_{1441}(381,\cdot)\) \(\chi_{1441}(382,\cdot)\) \(\chi_{1441}(475,\cdot)\) \(\chi_{1441}(590,\cdot)\) \(\chi_{1441}(591,\cdot)\) \(\chi_{1441}(596,\cdot)\) \(\chi_{1441}(611,\cdot)\) \(\chi_{1441}(612,\cdot)\) \(\chi_{1441}(622,\cdot)\) \(\chi_{1441}(651,\cdot)\) \(\chi_{1441}(657,\cdot)\) \(\chi_{1441}(711,\cdot)\) \(\chi_{1441}(745,\cdot)\) \(\chi_{1441}(761,\cdot)\) \(\chi_{1441}(765,\cdot)\) \(\chi_{1441}(777,\cdot)\) \(\chi_{1441}(783,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{43}{130}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{3}{13}\right)\)\(e\left(\frac{1}{65}\right)\)\(e\left(\frac{6}{13}\right)\)\(e\left(\frac{53}{65}\right)\)\(e\left(\frac{16}{65}\right)\)\(e\left(\frac{7}{130}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{2}{65}\right)\)\(e\left(\frac{3}{65}\right)\)\(e\left(\frac{31}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(17,a) \;\) at \(\;a = \) e.g. 2