Properties

Label 1441.1097
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([39,62]))
 
pari: [g,chi] = znchar(Mod(1097,1441))
 

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.bu

\(\chi_{1441}(46,\cdot)\) \(\chi_{1441}(74,\cdot)\) \(\chi_{1441}(94,\cdot)\) \(\chi_{1441}(101,\cdot)\) \(\chi_{1441}(138,\cdot)\) \(\chi_{1441}(222,\cdot)\) \(\chi_{1441}(282,\cdot)\) \(\chi_{1441}(303,\cdot)\) \(\chi_{1441}(370,\cdot)\) \(\chi_{1441}(371,\cdot)\) \(\chi_{1441}(391,\cdot)\) \(\chi_{1441}(409,\cdot)\) \(\chi_{1441}(414,\cdot)\) \(\chi_{1441}(448,\cdot)\) \(\chi_{1441}(457,\cdot)\) \(\chi_{1441}(470,\cdot)\) \(\chi_{1441}(514,\cdot)\) \(\chi_{1441}(557,\cdot)\) \(\chi_{1441}(568,\cdot)\) \(\chi_{1441}(629,\cdot)\) \(\chi_{1441}(666,\cdot)\) \(\chi_{1441}(690,\cdot)\) \(\chi_{1441}(789,\cdot)\) \(\chi_{1441}(799,\cdot)\) \(\chi_{1441}(811,\cdot)\) \(\chi_{1441}(909,\cdot)\) \(\chi_{1441}(921,\cdot)\) \(\chi_{1441}(926,\cdot)\) \(\chi_{1441}(976,\cdot)\) \(\chi_{1441}(992,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{31}{65}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(1097, a) \) \(-1\)\(1\)\(e\left(\frac{101}{130}\right)\)\(e\left(\frac{48}{65}\right)\)\(e\left(\frac{36}{65}\right)\)\(e\left(\frac{9}{65}\right)\)\(e\left(\frac{67}{130}\right)\)\(e\left(\frac{23}{26}\right)\)\(e\left(\frac{43}{130}\right)\)\(e\left(\frac{31}{65}\right)\)\(e\left(\frac{119}{130}\right)\)\(e\left(\frac{19}{65}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(1097,a) \;\) at \(\;a = \) e.g. 2