Properties

Label 1441.560
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1441, base_ring=CyclotomicField(130))
 
M = H._module
 
chi = DirichletCharacter(H, M([65,16]))
 
pari: [g,chi] = znchar(Mod(560,1441))
 

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

Galois orbit 1441.bq

\(\chi_{1441}(21,\cdot)\) \(\chi_{1441}(43,\cdot)\) \(\chi_{1441}(65,\cdot)\) \(\chi_{1441}(109,\cdot)\) \(\chi_{1441}(142,\cdot)\) \(\chi_{1441}(164,\cdot)\) \(\chi_{1441}(175,\cdot)\) \(\chi_{1441}(186,\cdot)\) \(\chi_{1441}(208,\cdot)\) \(\chi_{1441}(252,\cdot)\) \(\chi_{1441}(274,\cdot)\) \(\chi_{1441}(296,\cdot)\) \(\chi_{1441}(362,\cdot)\) \(\chi_{1441}(406,\cdot)\) \(\chi_{1441}(428,\cdot)\) \(\chi_{1441}(439,\cdot)\) \(\chi_{1441}(494,\cdot)\) \(\chi_{1441}(516,\cdot)\) \(\chi_{1441}(527,\cdot)\) \(\chi_{1441}(549,\cdot)\) \(\chi_{1441}(560,\cdot)\) \(\chi_{1441}(615,\cdot)\) \(\chi_{1441}(626,\cdot)\) \(\chi_{1441}(659,\cdot)\) \(\chi_{1441}(670,\cdot)\) \(\chi_{1441}(703,\cdot)\) \(\chi_{1441}(714,\cdot)\) \(\chi_{1441}(736,\cdot)\) \(\chi_{1441}(769,\cdot)\) \(\chi_{1441}(780,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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

\((1311,133)\) → \((-1,e\left(\frac{8}{65}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(560, a) \) \(-1\)\(1\)\(e\left(\frac{81}{130}\right)\)\(e\left(\frac{56}{65}\right)\)\(e\left(\frac{16}{65}\right)\)\(e\left(\frac{43}{65}\right)\)\(e\left(\frac{63}{130}\right)\)\(e\left(\frac{41}{130}\right)\)\(e\left(\frac{113}{130}\right)\)\(e\left(\frac{47}{65}\right)\)\(e\left(\frac{37}{130}\right)\)\(e\left(\frac{7}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(560,a) \;\) at \(\;a = \) e.g. 2