Properties

Label 61.38
Modulus $61$
Conductor $61$
Order $20$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([9]))
 
pari: [g,chi] = znchar(Mod(38,61))
 

Basic properties

Modulus: \(61\)
Conductor: \(61\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 61.j

\(\chi_{61}(8,\cdot)\) \(\chi_{61}(23,\cdot)\) \(\chi_{61}(24,\cdot)\) \(\chi_{61}(28,\cdot)\) \(\chi_{61}(33,\cdot)\) \(\chi_{61}(37,\cdot)\) \(\chi_{61}(38,\cdot)\) \(\chi_{61}(53,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: Number field defined by a degree 20 polynomial

Values on generators

\(2\) → \(e\left(\frac{9}{20}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 61 }(38, a) \) \(-1\)\(1\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 61 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 61 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 61 }(38,·),\chi_{ 61 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 61 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2