Properties

Label 1045.227
Modulus $1045$
Conductor $1045$
Order $20$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,14,10]))
 
pari: [g,chi] = znchar(Mod(227,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 1045.br

\(\chi_{1045}(18,\cdot)\) \(\chi_{1045}(227,\cdot)\) \(\chi_{1045}(303,\cdot)\) \(\chi_{1045}(398,\cdot)\) \(\chi_{1045}(512,\cdot)\) \(\chi_{1045}(607,\cdot)\) \(\chi_{1045}(778,\cdot)\) \(\chi_{1045}(987,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\((837,761,496)\) → \((i,e\left(\frac{7}{10}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(227, a) \) \(-1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(-i\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{3}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(227,a) \;\) at \(\;a = \) e.g. 2