Properties

Label 1045.414
Modulus $1045$
Conductor $1045$
Order $90$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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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([45,63,55]))
 
pari: [g,chi] = znchar(Mod(414,1045))
 

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.cn

\(\chi_{1045}(29,\cdot)\) \(\chi_{1045}(79,\cdot)\) \(\chi_{1045}(129,\cdot)\) \(\chi_{1045}(184,\cdot)\) \(\chi_{1045}(204,\cdot)\) \(\chi_{1045}(249,\cdot)\) \(\chi_{1045}(299,\cdot)\) \(\chi_{1045}(314,\cdot)\) \(\chi_{1045}(409,\cdot)\) \(\chi_{1045}(414,\cdot)\) \(\chi_{1045}(459,\cdot)\) \(\chi_{1045}(469,\cdot)\) \(\chi_{1045}(534,\cdot)\) \(\chi_{1045}(629,\cdot)\) \(\chi_{1045}(679,\cdot)\) \(\chi_{1045}(699,\cdot)\) \(\chi_{1045}(744,\cdot)\) \(\chi_{1045}(754,\cdot)\) \(\chi_{1045}(789,\cdot)\) \(\chi_{1045}(794,\cdot)\) \(\chi_{1045}(849,\cdot)\) \(\chi_{1045}(964,\cdot)\) \(\chi_{1045}(1009,\cdot)\) \(\chi_{1045}(1029,\cdot)\)

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

\((837,761,496)\) → \((-1,e\left(\frac{7}{10}\right),e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(414, a) \) \(1\)\(1\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{2}{45}\right)\)\(e\left(\frac{28}{45}\right)\)\(e\left(\frac{77}{90}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{4}{45}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{23}{90}\right)\)\(e\left(\frac{79}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(414,a) \;\) at \(\;a = \) e.g. 2