Properties

Label 1045.214
Modulus $1045$
Conductor $1045$
Order $90$
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(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,36,80]))
 
pari: [g,chi] = znchar(Mod(214,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
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.ck

\(\chi_{1045}(4,\cdot)\) \(\chi_{1045}(9,\cdot)\) \(\chi_{1045}(104,\cdot)\) \(\chi_{1045}(119,\cdot)\) \(\chi_{1045}(169,\cdot)\) \(\chi_{1045}(214,\cdot)\) \(\chi_{1045}(234,\cdot)\) \(\chi_{1045}(289,\cdot)\) \(\chi_{1045}(339,\cdot)\) \(\chi_{1045}(389,\cdot)\) \(\chi_{1045}(434,\cdot)\) \(\chi_{1045}(454,\cdot)\) \(\chi_{1045}(499,\cdot)\) \(\chi_{1045}(614,\cdot)\) \(\chi_{1045}(669,\cdot)\) \(\chi_{1045}(674,\cdot)\) \(\chi_{1045}(709,\cdot)\) \(\chi_{1045}(719,\cdot)\) \(\chi_{1045}(764,\cdot)\) \(\chi_{1045}(784,\cdot)\) \(\chi_{1045}(834,\cdot)\) \(\chi_{1045}(929,\cdot)\) \(\chi_{1045}(994,\cdot)\) \(\chi_{1045}(1004,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 90 polynomial

Values on generators

\((837,761,496)\) → \((-1,e\left(\frac{2}{5}\right),e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(214, a) \) \(1\)\(1\)\(e\left(\frac{71}{90}\right)\)\(e\left(\frac{23}{90}\right)\)\(e\left(\frac{26}{45}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{31}{90}\right)\)\(e\left(\frac{19}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(214,a) \;\) at \(\;a = \) e.g. 2