Properties

Label 1045.8
Modulus $1045$
Conductor $1045$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1045, base_ring=CyclotomicField(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([45,18,10]))
 
pari: [g,chi] = znchar(Mod(8,1045))
 

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

Galois orbit 1045.cg

\(\chi_{1045}(8,\cdot)\) \(\chi_{1045}(107,\cdot)\) \(\chi_{1045}(183,\cdot)\) \(\chi_{1045}(217,\cdot)\) \(\chi_{1045}(293,\cdot)\) \(\chi_{1045}(392,\cdot)\) \(\chi_{1045}(468,\cdot)\) \(\chi_{1045}(502,\cdot)\) \(\chi_{1045}(563,\cdot)\) \(\chi_{1045}(578,\cdot)\) \(\chi_{1045}(673,\cdot)\) \(\chi_{1045}(677,\cdot)\) \(\chi_{1045}(772,\cdot)\) \(\chi_{1045}(787,\cdot)\) \(\chi_{1045}(882,\cdot)\) \(\chi_{1045}(943,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((837,761,496)\) → \((-i,e\left(\frac{3}{10}\right),e\left(\frac{1}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{49}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{19}{30}\right)\)\(i\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(8,a) \;\) at \(\;a = \) e.g. 2