Properties

Label 1045.34
Modulus $1045$
Conductor $95$
Order $18$
Real no
Primitive no
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(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,0,11]))
 
pari: [g,chi] = znchar(Mod(34,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(95\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{95}(34,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1045.bl

\(\chi_{1045}(34,\cdot)\) \(\chi_{1045}(89,\cdot)\) \(\chi_{1045}(364,\cdot)\) \(\chi_{1045}(584,\cdot)\) \(\chi_{1045}(694,\cdot)\) \(\chi_{1045}(914,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.0.10703880581610941769412109375.1

Values on generators

\((837,761,496)\) → \((-1,1,e\left(\frac{11}{18}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(34,a) \;\) at \(\;a = \) e.g. 2