Properties

Label 95.53
Modulus $95$
Conductor $95$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(95, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([27,22]))
 
pari: [g,chi] = znchar(Mod(53,95))
 

Basic properties

Modulus: \(95\)
Conductor: \(95\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 95.r

\(\chi_{95}(2,\cdot)\) \(\chi_{95}(3,\cdot)\) \(\chi_{95}(13,\cdot)\) \(\chi_{95}(22,\cdot)\) \(\chi_{95}(32,\cdot)\) \(\chi_{95}(33,\cdot)\) \(\chi_{95}(48,\cdot)\) \(\chi_{95}(52,\cdot)\) \(\chi_{95}(53,\cdot)\) \(\chi_{95}(67,\cdot)\) \(\chi_{95}(72,\cdot)\) \(\chi_{95}(78,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{95})^+\)

Values on generators

\((77,21)\) → \((-i,e\left(\frac{11}{18}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 95 }(53, a) \) \(1\)\(1\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{36}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 95 }(53,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 95 }(53,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 95 }(53,·),\chi_{ 95 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 95 }(53,·)) \;\) at \(\; a,b = \) e.g. 1,2