Properties

Label 712.57
Modulus $712$
Conductor $89$
Order $22$
Real no
Primitive no
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(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,9]))
 
pari: [g,chi] = znchar(Mod(57,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(89\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{89}(57,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 712.u

\(\chi_{712}(25,\cdot)\) \(\chi_{712}(57,\cdot)\) \(\chi_{712}(73,\cdot)\) \(\chi_{712}(81,\cdot)\) \(\chi_{712}(265,\cdot)\) \(\chi_{712}(289,\cdot)\) \(\chi_{712}(441,\cdot)\) \(\chi_{712}(489,\cdot)\) \(\chi_{712}(545,\cdot)\) \(\chi_{712}(673,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: 22.22.86534669543385676516186776267386878120889.1

Values on generators

\((535,357,537)\) → \((1,1,e\left(\frac{9}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{6}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(57,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(57,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(57,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(57,·)) \;\) at \(\; a,b = \) e.g. 1,2