Properties

Label 1089.23
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([55,42]))
 
pari: [g,chi] = znchar(Mod(23,1089))
 

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.w

\(\chi_{1089}(23,\cdot)\) \(\chi_{1089}(56,\cdot)\) \(\chi_{1089}(155,\cdot)\) \(\chi_{1089}(221,\cdot)\) \(\chi_{1089}(254,\cdot)\) \(\chi_{1089}(320,\cdot)\) \(\chi_{1089}(353,\cdot)\) \(\chi_{1089}(419,\cdot)\) \(\chi_{1089}(452,\cdot)\) \(\chi_{1089}(518,\cdot)\) \(\chi_{1089}(551,\cdot)\) \(\chi_{1089}(617,\cdot)\) \(\chi_{1089}(650,\cdot)\) \(\chi_{1089}(716,\cdot)\) \(\chi_{1089}(749,\cdot)\) \(\chi_{1089}(815,\cdot)\) \(\chi_{1089}(914,\cdot)\) \(\chi_{1089}(947,\cdot)\) \(\chi_{1089}(1013,\cdot)\) \(\chi_{1089}(1046,\cdot)\)

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

\((848,244)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{7}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{31}{66}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(23,a) \;\) at \(\;a = \) e.g. 2