Properties

Label 712.211
Modulus $712$
Conductor $712$
Order $88$
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(712, base_ring=CyclotomicField(88))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,44,85]))
 
pari: [g,chi] = znchar(Mod(211,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(712\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
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 712.bf

\(\chi_{712}(3,\cdot)\) \(\chi_{712}(19,\cdot)\) \(\chi_{712}(27,\cdot)\) \(\chi_{712}(35,\cdot)\) \(\chi_{712}(43,\cdot)\) \(\chi_{712}(51,\cdot)\) \(\chi_{712}(59,\cdot)\) \(\chi_{712}(75,\cdot)\) \(\chi_{712}(83,\cdot)\) \(\chi_{712}(115,\cdot)\) \(\chi_{712}(147,\cdot)\) \(\chi_{712}(155,\cdot)\) \(\chi_{712}(163,\cdot)\) \(\chi_{712}(171,\cdot)\) \(\chi_{712}(211,\cdot)\) \(\chi_{712}(219,\cdot)\) \(\chi_{712}(243,\cdot)\) \(\chi_{712}(291,\cdot)\) \(\chi_{712}(315,\cdot)\) \(\chi_{712}(323,\cdot)\) \(\chi_{712}(363,\cdot)\) \(\chi_{712}(371,\cdot)\) \(\chi_{712}(379,\cdot)\) \(\chi_{712}(387,\cdot)\) \(\chi_{712}(419,\cdot)\) \(\chi_{712}(451,\cdot)\) \(\chi_{712}(459,\cdot)\) \(\chi_{712}(475,\cdot)\) \(\chi_{712}(483,\cdot)\) \(\chi_{712}(491,\cdot)\) ...

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: $\Q(\zeta_{88})$
Fixed field: Number field defined by a degree 88 polynomial

Values on generators

\((535,357,537)\) → \((-1,-1,e\left(\frac{85}{88}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{85}{88}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{65}{88}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{63}{88}\right)\)\(e\left(\frac{7}{88}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{71}{88}\right)\)\(e\left(\frac{31}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(211,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(211,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(211,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(211,·)) \;\) at \(\; a,b = \) e.g. 1,2