Properties

Label 1441.503
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([39,103]))
 
pari: [g,chi] = znchar(Mod(503,1441))
 

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.by

\(\chi_{1441}(2,\cdot)\) \(\chi_{1441}(8,\cdot)\) \(\chi_{1441}(90,\cdot)\) \(\chi_{1441}(106,\cdot)\) \(\chi_{1441}(118,\cdot)\) \(\chi_{1441}(128,\cdot)\) \(\chi_{1441}(171,\cdot)\) \(\chi_{1441}(227,\cdot)\) \(\chi_{1441}(255,\cdot)\) \(\chi_{1441}(292,\cdot)\) \(\chi_{1441}(299,\cdot)\) \(\chi_{1441}(347,\cdot)\) \(\chi_{1441}(349,\cdot)\) \(\chi_{1441}(360,\cdot)\) \(\chi_{1441}(403,\cdot)\) \(\chi_{1441}(424,\cdot)\) \(\chi_{1441}(447,\cdot)\) \(\chi_{1441}(469,\cdot)\) \(\chi_{1441}(486,\cdot)\) \(\chi_{1441}(490,\cdot)\) \(\chi_{1441}(497,\cdot)\) \(\chi_{1441}(503,\cdot)\) \(\chi_{1441}(512,\cdot)\) \(\chi_{1441}(519,\cdot)\) \(\chi_{1441}(546,\cdot)\) \(\chi_{1441}(547,\cdot)\) \(\chi_{1441}(607,\cdot)\) \(\chi_{1441}(635,\cdot)\) \(\chi_{1441}(684,\cdot)\) \(\chi_{1441}(721,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{103}{130}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{6}{65}\right)\)\(e\left(\frac{29}{65}\right)\)\(e\left(\frac{12}{65}\right)\)\(e\left(\frac{42}{65}\right)\)\(e\left(\frac{7}{13}\right)\)\(e\left(\frac{21}{130}\right)\)\(e\left(\frac{18}{65}\right)\)\(e\left(\frac{58}{65}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{41}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(503,a) \;\) at \(\;a = \) e.g. 2