Properties

Label 1045.747
Modulus $1045$
Conductor $1045$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,18,28]))
 
pari: [g,chi] = znchar(Mod(747,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
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 1045.cb

\(\chi_{1045}(43,\cdot)\) \(\chi_{1045}(142,\cdot)\) \(\chi_{1045}(252,\cdot)\) \(\chi_{1045}(263,\cdot)\) \(\chi_{1045}(472,\cdot)\) \(\chi_{1045}(538,\cdot)\) \(\chi_{1045}(593,\cdot)\) \(\chi_{1045}(747,\cdot)\) \(\chi_{1045}(758,\cdot)\) \(\chi_{1045}(802,\cdot)\) \(\chi_{1045}(967,\cdot)\) \(\chi_{1045}(978,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((837,761,496)\) → \((i,-1,e\left(\frac{7}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(747, a) \) \(1\)\(1\)\(e\left(\frac{19}{36}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{17}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(747,a) \;\) at \(\;a = \) e.g. 2