Properties

Label 1045.149
Modulus $1045$
Conductor $1045$
Order $90$
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(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,81,20]))
 
pari: [g,chi] = znchar(Mod(149,1045))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1045.cp

\(\chi_{1045}(24,\cdot)\) \(\chi_{1045}(74,\cdot)\) \(\chi_{1045}(139,\cdot)\) \(\chi_{1045}(149,\cdot)\) \(\chi_{1045}(194,\cdot)\) \(\chi_{1045}(244,\cdot)\) \(\chi_{1045}(294,\cdot)\) \(\chi_{1045}(359,\cdot)\) \(\chi_{1045}(404,\cdot)\) \(\chi_{1045}(424,\cdot)\) \(\chi_{1045}(479,\cdot)\) \(\chi_{1045}(519,\cdot)\) \(\chi_{1045}(574,\cdot)\) \(\chi_{1045}(579,\cdot)\) \(\chi_{1045}(624,\cdot)\) \(\chi_{1045}(644,\cdot)\) \(\chi_{1045}(689,\cdot)\) \(\chi_{1045}(739,\cdot)\) \(\chi_{1045}(864,\cdot)\) \(\chi_{1045}(899,\cdot)\) \(\chi_{1045}(909,\cdot)\) \(\chi_{1045}(954,\cdot)\) \(\chi_{1045}(959,\cdot)\) \(\chi_{1045}(974,\cdot)\)

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

\((837,761,496)\) → \((-1,e\left(\frac{9}{10}\right),e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(149, a) \) \(-1\)\(1\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{53}{90}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{19}{90}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{8}{45}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{34}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(149,a) \;\) at \(\;a = \) e.g. 2