Properties

Label 69.2
Modulus $69$
Conductor $69$
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(69, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,2]))
 
pari: [g,chi] = znchar(Mod(2,69))
 

Basic properties

Modulus: \(69\)
Conductor: \(69\)
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 69.h

\(\chi_{69}(2,\cdot)\) \(\chi_{69}(8,\cdot)\) \(\chi_{69}(26,\cdot)\) \(\chi_{69}(29,\cdot)\) \(\chi_{69}(32,\cdot)\) \(\chi_{69}(35,\cdot)\) \(\chi_{69}(41,\cdot)\) \(\chi_{69}(50,\cdot)\) \(\chi_{69}(59,\cdot)\) \(\chi_{69}(62,\cdot)\)

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: 22.0.304011857053427966889939263171547.1

Values on generators

\((47,28)\) → \((-1,e\left(\frac{1}{11}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(-1\)\(1\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{8}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 69 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 69 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 69 }(2,·),\chi_{ 69 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 69 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2