Properties

Label 712.331
Modulus $712$
Conductor $712$
Order $22$
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(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,11,2]))
 
pari: [g,chi] = znchar(Mod(331,712))
 

Basic properties

Modulus: \(712\)
Conductor: \(712\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
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.s

\(\chi_{712}(67,\cdot)\) \(\chi_{712}(91,\cdot)\) \(\chi_{712}(275,\cdot)\) \(\chi_{712}(283,\cdot)\) \(\chi_{712}(299,\cdot)\) \(\chi_{712}(331,\cdot)\) \(\chi_{712}(395,\cdot)\) \(\chi_{712}(523,\cdot)\) \(\chi_{712}(579,\cdot)\) \(\chi_{712}(627,\cdot)\)

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: 22.0.8351990464247387275322638622384170627163060436992.1

Values on generators

\((535,357,537)\) → \((-1,-1,e\left(\frac{1}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(-1\)\(1\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 712 }(331,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 712 }(331,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 712 }(331,·),\chi_{ 712 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 712 }(331,·)) \;\) at \(\; a,b = \) e.g. 1,2