Properties

Label 322.237
Modulus $322$
Conductor $161$
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(322, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,19]))
 
pari: [g,chi] = znchar(Mod(237,322))
 

Basic properties

Modulus: \(322\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(76,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 322.k

\(\chi_{322}(83,\cdot)\) \(\chi_{322}(97,\cdot)\) \(\chi_{322}(111,\cdot)\) \(\chi_{322}(125,\cdot)\) \(\chi_{322}(153,\cdot)\) \(\chi_{322}(181,\cdot)\) \(\chi_{322}(195,\cdot)\) \(\chi_{322}(237,\cdot)\) \(\chi_{322}(251,\cdot)\) \(\chi_{322}(293,\cdot)\)

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: 22.22.78048218870425324004237696277333187889.1

Values on generators

\((185,281)\) → \((-1,e\left(\frac{19}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(25\)\(27\)
\(1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 322 }(237,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 322 }(237,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 322 }(237,·),\chi_{ 322 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 322 }(237,·)) \;\) at \(\; a,b = \) e.g. 1,2