Properties

Label 712.281
Modulus $712$
Conductor $89$
Order $88$
Real no
Primitive no
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([0,0,9]))
 
pari: [g,chi] = znchar(Mod(281,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(89\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{89}(14,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 712.be

\(\chi_{712}(33,\cdot)\) \(\chi_{712}(41,\cdot)\) \(\chi_{712}(65,\cdot)\) \(\chi_{712}(113,\cdot)\) \(\chi_{712}(137,\cdot)\) \(\chi_{712}(145,\cdot)\) \(\chi_{712}(185,\cdot)\) \(\chi_{712}(193,\cdot)\) \(\chi_{712}(201,\cdot)\) \(\chi_{712}(209,\cdot)\) \(\chi_{712}(241,\cdot)\) \(\chi_{712}(273,\cdot)\) \(\chi_{712}(281,\cdot)\) \(\chi_{712}(297,\cdot)\) \(\chi_{712}(305,\cdot)\) \(\chi_{712}(313,\cdot)\) \(\chi_{712}(321,\cdot)\) \(\chi_{712}(329,\cdot)\) \(\chi_{712}(337,\cdot)\) \(\chi_{712}(353,\cdot)\) \(\chi_{712}(369,\cdot)\) \(\chi_{712}(385,\cdot)\) \(\chi_{712}(417,\cdot)\) \(\chi_{712}(473,\cdot)\) \(\chi_{712}(505,\cdot)\) \(\chi_{712}(521,\cdot)\) \(\chi_{712}(537,\cdot)\) \(\chi_{712}(553,\cdot)\) \(\chi_{712}(561,\cdot)\) \(\chi_{712}(569,\cdot)\) ...

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

\((535,357,537)\) → \((1,1,e\left(\frac{9}{88}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{9}{88}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{25}{88}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{31}{88}\right)\)\(e\left(\frac{23}{88}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{51}{88}\right)\)\(e\left(\frac{17}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(281,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(281,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(281,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(281,·)) \;\) at \(\; a,b = \) e.g. 1,2