Properties

Label 1089.41
Modulus $1089$
Conductor $1089$
Order $330$
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([275,69]))
 
pari: [g,chi] = znchar(Mod(41,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(1089\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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.bd

\(\chi_{1089}(2,\cdot)\) \(\chi_{1089}(29,\cdot)\) \(\chi_{1089}(41,\cdot)\) \(\chi_{1089}(50,\cdot)\) \(\chi_{1089}(68,\cdot)\) \(\chi_{1089}(74,\cdot)\) \(\chi_{1089}(83,\cdot)\) \(\chi_{1089}(95,\cdot)\) \(\chi_{1089}(101,\cdot)\) \(\chi_{1089}(128,\cdot)\) \(\chi_{1089}(140,\cdot)\) \(\chi_{1089}(149,\cdot)\) \(\chi_{1089}(167,\cdot)\) \(\chi_{1089}(173,\cdot)\) \(\chi_{1089}(182,\cdot)\) \(\chi_{1089}(194,\cdot)\) \(\chi_{1089}(200,\cdot)\) \(\chi_{1089}(227,\cdot)\) \(\chi_{1089}(248,\cdot)\) \(\chi_{1089}(266,\cdot)\) \(\chi_{1089}(272,\cdot)\) \(\chi_{1089}(281,\cdot)\) \(\chi_{1089}(293,\cdot)\) \(\chi_{1089}(299,\cdot)\) \(\chi_{1089}(326,\cdot)\) \(\chi_{1089}(338,\cdot)\) \(\chi_{1089}(347,\cdot)\) \(\chi_{1089}(365,\cdot)\) \(\chi_{1089}(371,\cdot)\) \(\chi_{1089}(380,\cdot)\) ...

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 330 polynomial (not computed)

Values on generators

\((848,244)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{23}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{7}{165}\right)\)\(e\left(\frac{14}{165}\right)\)\(e\left(\frac{211}{330}\right)\)\(e\left(\frac{263}{330}\right)\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{259}{330}\right)\)\(e\left(\frac{277}{330}\right)\)\(e\left(\frac{28}{165}\right)\)\(e\left(\frac{41}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(41,a) \;\) at \(\;a = \) e.g. 2