Properties

Label 1441.673
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
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([13,15]))
 
pari: [g,chi] = znchar(Mod(673,1441))
 

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.bz

\(\chi_{1441}(18,\cdot)\) \(\chi_{1441}(19,\cdot)\) \(\chi_{1441}(24,\cdot)\) \(\chi_{1441}(51,\cdot)\) \(\chi_{1441}(68,\cdot)\) \(\chi_{1441}(79,\cdot)\) \(\chi_{1441}(149,\cdot)\) \(\chi_{1441}(150,\cdot)\) \(\chi_{1441}(178,\cdot)\) \(\chi_{1441}(182,\cdot)\) \(\chi_{1441}(200,\cdot)\) \(\chi_{1441}(217,\cdot)\) \(\chi_{1441}(281,\cdot)\) \(\chi_{1441}(294,\cdot)\) \(\chi_{1441}(348,\cdot)\) \(\chi_{1441}(354,\cdot)\) \(\chi_{1441}(425,\cdot)\) \(\chi_{1441}(464,\cdot)\) \(\chi_{1441}(479,\cdot)\) \(\chi_{1441}(556,\cdot)\) \(\chi_{1441}(673,\cdot)\) \(\chi_{1441}(679,\cdot)\) \(\chi_{1441}(706,\cdot)\) \(\chi_{1441}(723,\cdot)\) \(\chi_{1441}(734,\cdot)\) \(\chi_{1441}(805,\cdot)\) \(\chi_{1441}(810,\cdot)\) \(\chi_{1441}(833,\cdot)\) \(\chi_{1441}(854,\cdot)\) \(\chi_{1441}(855,\cdot)\) ...

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

\((1311,133)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{3}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(673, a) \) \(1\)\(1\)\(e\left(\frac{14}{65}\right)\)\(e\left(\frac{7}{65}\right)\)\(e\left(\frac{28}{65}\right)\)\(e\left(\frac{46}{65}\right)\)\(e\left(\frac{21}{65}\right)\)\(e\left(\frac{101}{130}\right)\)\(e\left(\frac{42}{65}\right)\)\(e\left(\frac{14}{65}\right)\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{7}{13}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1441 }(673,a) \;\) at \(\;a = \) e.g. 2