Properties

Label 135.47
Modulus $135$
Conductor $135$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,9]))
 
pari: [g,chi] = znchar(Mod(47,135))
 

Basic properties

Modulus: \(135\)
Conductor: \(135\)
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 135.q

\(\chi_{135}(2,\cdot)\) \(\chi_{135}(23,\cdot)\) \(\chi_{135}(32,\cdot)\) \(\chi_{135}(38,\cdot)\) \(\chi_{135}(47,\cdot)\) \(\chi_{135}(68,\cdot)\) \(\chi_{135}(77,\cdot)\) \(\chi_{135}(83,\cdot)\) \(\chi_{135}(92,\cdot)\) \(\chi_{135}(113,\cdot)\) \(\chi_{135}(122,\cdot)\) \(\chi_{135}(128,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: \(\Q(\zeta_{135})^+\)

Values on generators

\((56,82)\) → \((e\left(\frac{7}{18}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 135 }(47, a) \) \(1\)\(1\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 135 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 135 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 135 }(47,·),\chi_{ 135 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 135 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2