Properties

Label 1089.43
Modulus $1089$
Conductor $1089$
Order $66$
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(1089, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,15]))
 
pari: [g,chi] = znchar(Mod(43,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(1089\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 1089.x

\(\chi_{1089}(43,\cdot)\) \(\chi_{1089}(76,\cdot)\) \(\chi_{1089}(142,\cdot)\) \(\chi_{1089}(175,\cdot)\) \(\chi_{1089}(274,\cdot)\) \(\chi_{1089}(340,\cdot)\) \(\chi_{1089}(373,\cdot)\) \(\chi_{1089}(439,\cdot)\) \(\chi_{1089}(472,\cdot)\) \(\chi_{1089}(538,\cdot)\) \(\chi_{1089}(571,\cdot)\) \(\chi_{1089}(637,\cdot)\) \(\chi_{1089}(670,\cdot)\) \(\chi_{1089}(736,\cdot)\) \(\chi_{1089}(769,\cdot)\) \(\chi_{1089}(835,\cdot)\) \(\chi_{1089}(868,\cdot)\) \(\chi_{1089}(934,\cdot)\) \(\chi_{1089}(1033,\cdot)\) \(\chi_{1089}(1066,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((848,244)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{5}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{3}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(43,a) \;\) at \(\;a = \) e.g. 2