Properties

Label 1089.322
Modulus $1089$
Conductor $1089$
Order $165$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1089, base_ring=CyclotomicField(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([220,234]))
 
pari: [g,chi] = znchar(Mod(322,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(1089\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(165\)
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 1089.bc

\(\chi_{1089}(4,\cdot)\) \(\chi_{1089}(16,\cdot)\) \(\chi_{1089}(25,\cdot)\) \(\chi_{1089}(31,\cdot)\) \(\chi_{1089}(49,\cdot)\) \(\chi_{1089}(58,\cdot)\) \(\chi_{1089}(70,\cdot)\) \(\chi_{1089}(97,\cdot)\) \(\chi_{1089}(103,\cdot)\) \(\chi_{1089}(115,\cdot)\) \(\chi_{1089}(157,\cdot)\) \(\chi_{1089}(169,\cdot)\) \(\chi_{1089}(196,\cdot)\) \(\chi_{1089}(214,\cdot)\) \(\chi_{1089}(223,\cdot)\) \(\chi_{1089}(229,\cdot)\) \(\chi_{1089}(247,\cdot)\) \(\chi_{1089}(256,\cdot)\) \(\chi_{1089}(268,\cdot)\) \(\chi_{1089}(295,\cdot)\) \(\chi_{1089}(301,\cdot)\) \(\chi_{1089}(313,\cdot)\) \(\chi_{1089}(322,\cdot)\) \(\chi_{1089}(328,\cdot)\) \(\chi_{1089}(346,\cdot)\) \(\chi_{1089}(355,\cdot)\) \(\chi_{1089}(367,\cdot)\) \(\chi_{1089}(394,\cdot)\) \(\chi_{1089}(400,\cdot)\) \(\chi_{1089}(412,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 165 polynomial (not computed)

Values on generators

\((848,244)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{39}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{62}{165}\right)\)\(e\left(\frac{124}{165}\right)\)\(e\left(\frac{133}{165}\right)\)\(e\left(\frac{104}{165}\right)\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{157}{165}\right)\)\(e\left(\frac{1}{165}\right)\)\(e\left(\frac{83}{165}\right)\)\(e\left(\frac{41}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(322,a) \;\) at \(\;a = \) e.g. 2