Properties

Label 1045.2
Modulus $1045$
Conductor $1045$
Order $180$
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(180))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([45,18,10]))
 
pari: [g,chi] = znchar(Mod(2,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(1045\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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.cq

\(\chi_{1045}(2,\cdot)\) \(\chi_{1045}(13,\cdot)\) \(\chi_{1045}(52,\cdot)\) \(\chi_{1045}(72,\cdot)\) \(\chi_{1045}(117,\cdot)\) \(\chi_{1045}(127,\cdot)\) \(\chi_{1045}(128,\cdot)\) \(\chi_{1045}(162,\cdot)\) \(\chi_{1045}(167,\cdot)\) \(\chi_{1045}(173,\cdot)\) \(\chi_{1045}(193,\cdot)\) \(\chi_{1045}(222,\cdot)\) \(\chi_{1045}(238,\cdot)\) \(\chi_{1045}(288,\cdot)\) \(\chi_{1045}(337,\cdot)\) \(\chi_{1045}(338,\cdot)\) \(\chi_{1045}(382,\cdot)\) \(\chi_{1045}(393,\cdot)\) \(\chi_{1045}(402,\cdot)\) \(\chi_{1045}(413,\cdot)\) \(\chi_{1045}(447,\cdot)\) \(\chi_{1045}(458,\cdot)\) \(\chi_{1045}(497,\cdot)\) \(\chi_{1045}(508,\cdot)\) \(\chi_{1045}(523,\cdot)\) \(\chi_{1045}(547,\cdot)\) \(\chi_{1045}(602,\cdot)\) \(\chi_{1045}(618,\cdot)\) \(\chi_{1045}(622,\cdot)\) \(\chi_{1045}(623,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

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

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{73}{180}\right)\)\(e\left(\frac{49}{180}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{180}\right)\)\(e\left(\frac{31}{45}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(2,a) \;\) at \(\;a = \) e.g. 2