Properties

Label 1045.657
Modulus $1045$
Conductor $1045$
Order $60$
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(60))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,18,40]))
 
pari: [g,chi] = znchar(Mod(657,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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.cf

\(\chi_{1045}(7,\cdot)\) \(\chi_{1045}(68,\cdot)\) \(\chi_{1045}(83,\cdot)\) \(\chi_{1045}(178,\cdot)\) \(\chi_{1045}(182,\cdot)\) \(\chi_{1045}(277,\cdot)\) \(\chi_{1045}(292,\cdot)\) \(\chi_{1045}(387,\cdot)\) \(\chi_{1045}(448,\cdot)\) \(\chi_{1045}(558,\cdot)\) \(\chi_{1045}(657,\cdot)\) \(\chi_{1045}(733,\cdot)\) \(\chi_{1045}(767,\cdot)\) \(\chi_{1045}(843,\cdot)\) \(\chi_{1045}(942,\cdot)\) \(\chi_{1045}(1018,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((837,761,496)\) → \((i,e\left(\frac{3}{10}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(657, a) \) \(1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(i\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{17}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(657,a) \;\) at \(\;a = \) e.g. 2