Properties

Label 712.71
Modulus $712$
Conductor $356$
Order $44$
Real no
Primitive no
Minimal no
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(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([22,0,31]))
 
pari: [g,chi] = znchar(Mod(71,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(356\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{356}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 712.bb

\(\chi_{712}(47,\cdot)\) \(\chi_{712}(71,\cdot)\) \(\chi_{712}(79,\cdot)\) \(\chi_{712}(183,\cdot)\) \(\chi_{712}(199,\cdot)\) \(\chi_{712}(231,\cdot)\) \(\chi_{712}(247,\cdot)\) \(\chi_{712}(287,\cdot)\) \(\chi_{712}(303,\cdot)\) \(\chi_{712}(335,\cdot)\) \(\chi_{712}(351,\cdot)\) \(\chi_{712}(455,\cdot)\) \(\chi_{712}(463,\cdot)\) \(\chi_{712}(487,\cdot)\) \(\chi_{712}(543,\cdot)\) \(\chi_{712}(551,\cdot)\) \(\chi_{712}(583,\cdot)\) \(\chi_{712}(663,\cdot)\) \(\chi_{712}(695,\cdot)\) \(\chi_{712}(703,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.0.11724385642028745774656376755923007752612263949364698259094951647151017819768316744951538808520704.1

Values on generators

\((535,357,537)\) → \((-1,1,e\left(\frac{31}{44}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{23}{44}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{17}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(71,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(71,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(71,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(71,·)) \;\) at \(\; a,b = \) e.g. 1,2