Properties

Label 712.7
Modulus $712$
Conductor $356$
Order $88$
Real no
Primitive no
Minimal no
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([44,0,81]))
 
pari: [g,chi] = znchar(Mod(7,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(356\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(88\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{356}(7,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 712.bd

\(\chi_{712}(7,\cdot)\) \(\chi_{712}(15,\cdot)\) \(\chi_{712}(23,\cdot)\) \(\chi_{712}(31,\cdot)\) \(\chi_{712}(63,\cdot)\) \(\chi_{712}(95,\cdot)\) \(\chi_{712}(103,\cdot)\) \(\chi_{712}(119,\cdot)\) \(\chi_{712}(127,\cdot)\) \(\chi_{712}(135,\cdot)\) \(\chi_{712}(143,\cdot)\) \(\chi_{712}(151,\cdot)\) \(\chi_{712}(159,\cdot)\) \(\chi_{712}(175,\cdot)\) \(\chi_{712}(191,\cdot)\) \(\chi_{712}(207,\cdot)\) \(\chi_{712}(239,\cdot)\) \(\chi_{712}(295,\cdot)\) \(\chi_{712}(327,\cdot)\) \(\chi_{712}(343,\cdot)\) \(\chi_{712}(359,\cdot)\) \(\chi_{712}(375,\cdot)\) \(\chi_{712}(383,\cdot)\) \(\chi_{712}(391,\cdot)\) \(\chi_{712}(399,\cdot)\) \(\chi_{712}(407,\cdot)\) \(\chi_{712}(415,\cdot)\) \(\chi_{712}(431,\cdot)\) \(\chi_{712}(439,\cdot)\) \(\chi_{712}(471,\cdot)\) ...

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

\((535,357,537)\) → \((-1,1,e\left(\frac{81}{88}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{37}{88}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{5}{88}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{15}{88}\right)\)\(e\left(\frac{75}{88}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{63}{88}\right)\)\(e\left(\frac{21}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(7,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(7,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(7,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(7,·)) \;\) at \(\; a,b = \) e.g. 1,2