Properties

Label 138.113
Modulus $138$
Conductor $69$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(138, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,13]))
 
pari: [g,chi] = znchar(Mod(113,138))
 

Basic properties

Modulus: \(138\)
Conductor: \(69\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{69}(44,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 138.f

\(\chi_{138}(5,\cdot)\) \(\chi_{138}(11,\cdot)\) \(\chi_{138}(17,\cdot)\) \(\chi_{138}(53,\cdot)\) \(\chi_{138}(65,\cdot)\) \(\chi_{138}(83,\cdot)\) \(\chi_{138}(89,\cdot)\) \(\chi_{138}(107,\cdot)\) \(\chi_{138}(113,\cdot)\) \(\chi_{138}(125,\cdot)\)

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: \(\Q(\zeta_{69})^+\)

Values on generators

\((47,97)\) → \((-1,e\left(\frac{13}{22}\right))\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 138 }(113, a) \) \(1\)\(1\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 138 }(113,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 138 }(113,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 138 }(113,·),\chi_{ 138 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 138 }(113,·)) \;\) at \(\; a,b = \) e.g. 1,2