Properties

Label 1441.551
Modulus $1441$
Conductor $131$
Order $65$
Real no
Primitive no
Minimal yes
Parity even

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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([0,86]))
 
pari: [g,chi] = znchar(Mod(551,1441))
 

Basic properties

Modulus: \(1441\)
Conductor: \(131\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(65\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{131}(27,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1441.bl

\(\chi_{1441}(12,\cdot)\) \(\chi_{1441}(34,\cdot)\) \(\chi_{1441}(100,\cdot)\) \(\chi_{1441}(144,\cdot)\) \(\chi_{1441}(166,\cdot)\) \(\chi_{1441}(177,\cdot)\) \(\chi_{1441}(232,\cdot)\) \(\chi_{1441}(254,\cdot)\) \(\chi_{1441}(265,\cdot)\) \(\chi_{1441}(287,\cdot)\) \(\chi_{1441}(298,\cdot)\) \(\chi_{1441}(353,\cdot)\) \(\chi_{1441}(364,\cdot)\) \(\chi_{1441}(397,\cdot)\) \(\chi_{1441}(408,\cdot)\) \(\chi_{1441}(441,\cdot)\) \(\chi_{1441}(452,\cdot)\) \(\chi_{1441}(474,\cdot)\) \(\chi_{1441}(507,\cdot)\) \(\chi_{1441}(518,\cdot)\) \(\chi_{1441}(529,\cdot)\) \(\chi_{1441}(540,\cdot)\) \(\chi_{1441}(551,\cdot)\) \(\chi_{1441}(562,\cdot)\) \(\chi_{1441}(573,\cdot)\) \(\chi_{1441}(683,\cdot)\) \(\chi_{1441}(749,\cdot)\) \(\chi_{1441}(760,\cdot)\) \(\chi_{1441}(793,\cdot)\) \(\chi_{1441}(903,\cdot)\) ...

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 65 polynomial

Values on generators

\((1311,133)\) → \((1,e\left(\frac{43}{65}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(551, a) \) \(1\)\(1\)\(e\left(\frac{43}{65}\right)\)\(e\left(\frac{41}{65}\right)\)\(e\left(\frac{21}{65}\right)\)\(e\left(\frac{28}{65}\right)\)\(e\left(\frac{19}{65}\right)\)\(e\left(\frac{33}{65}\right)\)\(e\left(\frac{64}{65}\right)\)\(e\left(\frac{17}{65}\right)\)\(e\left(\frac{6}{65}\right)\)\(e\left(\frac{62}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(551,a) \;\) at \(\;a = \) e.g. 2