Properties

Label 1045.287
Modulus $1045$
Conductor $95$
Order $36$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,0,2]))
 
pari: [g,chi] = znchar(Mod(287,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(95\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{95}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1045.cc

\(\chi_{1045}(67,\cdot)\) \(\chi_{1045}(78,\cdot)\) \(\chi_{1045}(243,\cdot)\) \(\chi_{1045}(287,\cdot)\) \(\chi_{1045}(298,\cdot)\) \(\chi_{1045}(452,\cdot)\) \(\chi_{1045}(507,\cdot)\) \(\chi_{1045}(573,\cdot)\) \(\chi_{1045}(782,\cdot)\) \(\chi_{1045}(793,\cdot)\) \(\chi_{1045}(903,\cdot)\) \(\chi_{1045}(1002,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{95})^+\)

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 1045 }(287, a) \) \(1\)\(1\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(287,a) \;\) at \(\;a = \) e.g. 2