Properties

Label 1089.38
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([55,252]))
 
pari: [g,chi] = znchar(Mod(38,1089))
 

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.bf

\(\chi_{1089}(5,\cdot)\) \(\chi_{1089}(14,\cdot)\) \(\chi_{1089}(20,\cdot)\) \(\chi_{1089}(38,\cdot)\) \(\chi_{1089}(47,\cdot)\) \(\chi_{1089}(59,\cdot)\) \(\chi_{1089}(86,\cdot)\) \(\chi_{1089}(92,\cdot)\) \(\chi_{1089}(104,\cdot)\) \(\chi_{1089}(113,\cdot)\) \(\chi_{1089}(119,\cdot)\) \(\chi_{1089}(137,\cdot)\) \(\chi_{1089}(146,\cdot)\) \(\chi_{1089}(158,\cdot)\) \(\chi_{1089}(185,\cdot)\) \(\chi_{1089}(191,\cdot)\) \(\chi_{1089}(203,\cdot)\) \(\chi_{1089}(212,\cdot)\) \(\chi_{1089}(218,\cdot)\) \(\chi_{1089}(236,\cdot)\) \(\chi_{1089}(257,\cdot)\) \(\chi_{1089}(284,\cdot)\) \(\chi_{1089}(290,\cdot)\) \(\chi_{1089}(302,\cdot)\) \(\chi_{1089}(311,\cdot)\) \(\chi_{1089}(317,\cdot)\) \(\chi_{1089}(335,\cdot)\) \(\chi_{1089}(344,\cdot)\) \(\chi_{1089}(356,\cdot)\) \(\chi_{1089}(383,\cdot)\) ...

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

\((848,244)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{42}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{307}{330}\right)\)\(e\left(\frac{142}{165}\right)\)\(e\left(\frac{113}{330}\right)\)\(e\left(\frac{2}{165}\right)\)\(e\left(\frac{87}{110}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{76}{165}\right)\)\(e\left(\frac{311}{330}\right)\)\(e\left(\frac{119}{165}\right)\)\(e\left(\frac{101}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(38,a) \;\) at \(\;a = \) e.g. 2