Properties

Label 536.155
Modulus $536$
Conductor $536$
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(536, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,33,62]))
 
pari: [g,chi] = znchar(Mod(155,536))
 

Basic properties

Modulus: \(536\)
Conductor: \(536\)
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 536.ba

\(\chi_{536}(19,\cdot)\) \(\chi_{536}(35,\cdot)\) \(\chi_{536}(83,\cdot)\) \(\chi_{536}(123,\cdot)\) \(\chi_{536}(155,\cdot)\) \(\chi_{536}(211,\cdot)\) \(\chi_{536}(227,\cdot)\) \(\chi_{536}(291,\cdot)\) \(\chi_{536}(307,\cdot)\) \(\chi_{536}(315,\cdot)\) \(\chi_{536}(323,\cdot)\) \(\chi_{536}(339,\cdot)\) \(\chi_{536}(371,\cdot)\) \(\chi_{536}(395,\cdot)\) \(\chi_{536}(419,\cdot)\) \(\chi_{536}(435,\cdot)\) \(\chi_{536}(451,\cdot)\) \(\chi_{536}(467,\cdot)\) \(\chi_{536}(475,\cdot)\) \(\chi_{536}(523,\cdot)\)

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

\((135,269,337)\) → \((-1,-1,e\left(\frac{31}{33}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{7}{66}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{4}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{49}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 536 }(155,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 536 }(155,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 536 }(155,·),\chi_{ 536 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 536 }(155,·)) \;\) at \(\; a,b = \) e.g. 1,2