Properties

Label 62.9
Modulus $62$
Conductor $31$
Order $15$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(62, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([2]))
 
pari: [g,chi] = znchar(Mod(9,62))
 

Basic properties

Modulus: \(62\)
Conductor: \(31\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(15\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{31}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 62.g

\(\chi_{62}(7,\cdot)\) \(\chi_{62}(9,\cdot)\) \(\chi_{62}(19,\cdot)\) \(\chi_{62}(41,\cdot)\) \(\chi_{62}(45,\cdot)\) \(\chi_{62}(49,\cdot)\) \(\chi_{62}(51,\cdot)\) \(\chi_{62}(59,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 15 polynomial

Values on generators

\(3\) → \(e\left(\frac{1}{15}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 62 }(9, a) \) \(1\)\(1\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{14}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 62 }(9,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 62 }(9,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 62 }(9,·),\chi_{ 62 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 62 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2