Properties

Label 712.165
Modulus $712$
Conductor $712$
Order $88$
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(712, base_ring=CyclotomicField(88))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,44,67]))
 
pari: [g,chi] = znchar(Mod(165,712))
 

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

Galois orbit 712.bc

\(\chi_{712}(13,\cdot)\) \(\chi_{712}(29,\cdot)\) \(\chi_{712}(61,\cdot)\) \(\chi_{712}(117,\cdot)\) \(\chi_{712}(149,\cdot)\) \(\chi_{712}(165,\cdot)\) \(\chi_{712}(181,\cdot)\) \(\chi_{712}(197,\cdot)\) \(\chi_{712}(205,\cdot)\) \(\chi_{712}(213,\cdot)\) \(\chi_{712}(221,\cdot)\) \(\chi_{712}(229,\cdot)\) \(\chi_{712}(237,\cdot)\) \(\chi_{712}(253,\cdot)\) \(\chi_{712}(261,\cdot)\) \(\chi_{712}(293,\cdot)\) \(\chi_{712}(325,\cdot)\) \(\chi_{712}(333,\cdot)\) \(\chi_{712}(341,\cdot)\) \(\chi_{712}(349,\cdot)\) \(\chi_{712}(389,\cdot)\) \(\chi_{712}(397,\cdot)\) \(\chi_{712}(421,\cdot)\) \(\chi_{712}(469,\cdot)\) \(\chi_{712}(493,\cdot)\) \(\chi_{712}(501,\cdot)\) \(\chi_{712}(541,\cdot)\) \(\chi_{712}(549,\cdot)\) \(\chi_{712}(557,\cdot)\) \(\chi_{712}(565,\cdot)\) ...

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

\((535,357,537)\) → \((1,-1,e\left(\frac{67}{88}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{23}{88}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{59}{88}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{1}{88}\right)\)\(e\left(\frac{5}{88}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{13}{88}\right)\)\(e\left(\frac{41}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(165,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(165,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(165,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(165,·)) \;\) at \(\; a,b = \) e.g. 1,2