Properties

Label 1045.112
Modulus $1045$
Conductor $1045$
Order $180$
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(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([45,18,100]))
 
pari: [g,chi] = znchar(Mod(112,1045))
 

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

Galois orbit 1045.cs

\(\chi_{1045}(17,\cdot)\) \(\chi_{1045}(28,\cdot)\) \(\chi_{1045}(62,\cdot)\) \(\chi_{1045}(63,\cdot)\) \(\chi_{1045}(73,\cdot)\) \(\chi_{1045}(112,\cdot)\) \(\chi_{1045}(118,\cdot)\) \(\chi_{1045}(123,\cdot)\) \(\chi_{1045}(138,\cdot)\) \(\chi_{1045}(233,\cdot)\) \(\chi_{1045}(237,\cdot)\) \(\chi_{1045}(272,\cdot)\) \(\chi_{1045}(282,\cdot)\) \(\chi_{1045}(283,\cdot)\) \(\chi_{1045}(327,\cdot)\) \(\chi_{1045}(332,\cdot)\) \(\chi_{1045}(347,\cdot)\) \(\chi_{1045}(348,\cdot)\) \(\chi_{1045}(358,\cdot)\) \(\chi_{1045}(403,\cdot)\) \(\chi_{1045}(442,\cdot)\) \(\chi_{1045}(453,\cdot)\) \(\chi_{1045}(492,\cdot)\) \(\chi_{1045}(503,\cdot)\) \(\chi_{1045}(557,\cdot)\) \(\chi_{1045}(567,\cdot)\) \(\chi_{1045}(568,\cdot)\) \(\chi_{1045}(612,\cdot)\) \(\chi_{1045}(613,\cdot)\) \(\chi_{1045}(633,\cdot)\) ...

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

Values on generators

\((837,761,496)\) → \((i,e\left(\frac{1}{10}\right),e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(112, a) \) \(1\)\(1\)\(e\left(\frac{163}{180}\right)\)\(e\left(\frac{139}{180}\right)\)\(e\left(\frac{73}{90}\right)\)\(e\left(\frac{61}{90}\right)\)\(e\left(\frac{17}{60}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{49}{90}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{113}{180}\right)\)\(e\left(\frac{17}{90}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(112,a) \;\) at \(\;a = \) e.g. 2