Properties

Label 1064.25
Modulus $1064$
Conductor $133$
Order $9$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1064, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,12,14]))
 
pari: [g,chi] = znchar(Mod(25,1064))
 

Basic properties

Modulus: \(1064\)
Conductor: \(133\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{133}(25,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1064.de

\(\chi_{1064}(25,\cdot)\) \(\chi_{1064}(137,\cdot)\) \(\chi_{1064}(233,\cdot)\) \(\chi_{1064}(625,\cdot)\) \(\chi_{1064}(681,\cdot)\) \(\chi_{1064}(921,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 9.9.1998099208210609.1

Values on generators

\((799,533,913,1009)\) → \((1,1,e\left(\frac{2}{3}\right),e\left(\frac{7}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(23\)\(25\)\(27\)
\( \chi_{ 1064 }(25, a) \) \(1\)\(1\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1064 }(25,a) \;\) at \(\;a = \) e.g. 2