Properties

Label 1089.85
Modulus $1089$
Conductor $1089$
Order $330$
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(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([110,39]))
 
pari: [g,chi] = znchar(Mod(85,1089))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.be

\(\chi_{1089}(7,\cdot)\) \(\chi_{1089}(13,\cdot)\) \(\chi_{1089}(52,\cdot)\) \(\chi_{1089}(61,\cdot)\) \(\chi_{1089}(79,\cdot)\) \(\chi_{1089}(85,\cdot)\) \(\chi_{1089}(106,\cdot)\) \(\chi_{1089}(139,\cdot)\) \(\chi_{1089}(151,\cdot)\) \(\chi_{1089}(160,\cdot)\) \(\chi_{1089}(178,\cdot)\) \(\chi_{1089}(184,\cdot)\) \(\chi_{1089}(193,\cdot)\) \(\chi_{1089}(205,\cdot)\) \(\chi_{1089}(211,\cdot)\) \(\chi_{1089}(238,\cdot)\) \(\chi_{1089}(250,\cdot)\) \(\chi_{1089}(259,\cdot)\) \(\chi_{1089}(277,\cdot)\) \(\chi_{1089}(283,\cdot)\) \(\chi_{1089}(292,\cdot)\) \(\chi_{1089}(304,\cdot)\) \(\chi_{1089}(310,\cdot)\) \(\chi_{1089}(337,\cdot)\) \(\chi_{1089}(349,\cdot)\) \(\chi_{1089}(358,\cdot)\) \(\chi_{1089}(376,\cdot)\) \(\chi_{1089}(382,\cdot)\) \(\chi_{1089}(391,\cdot)\) \(\chi_{1089}(409,\cdot)\) ...

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

\((848,244)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{13}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{149}{330}\right)\)\(e\left(\frac{149}{165}\right)\)\(e\left(\frac{68}{165}\right)\)\(e\left(\frac{53}{330}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{199}{330}\right)\)\(e\left(\frac{101}{165}\right)\)\(e\left(\frac{133}{165}\right)\)\(e\left(\frac{87}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(85,a) \;\) at \(\;a = \) e.g. 2