Properties

Label 55.13
Modulus $55$
Conductor $55$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,2]))
 
pari: [g,chi] = znchar(Mod(13,55))
 

Basic properties

Modulus: \(55\)
Conductor: \(55\)
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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 55.l

\(\chi_{55}(2,\cdot)\) \(\chi_{55}(7,\cdot)\) \(\chi_{55}(8,\cdot)\) \(\chi_{55}(13,\cdot)\) \(\chi_{55}(17,\cdot)\) \(\chi_{55}(18,\cdot)\) \(\chi_{55}(28,\cdot)\) \(\chi_{55}(52,\cdot)\)

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: \(\Q(\zeta_{55})^+\)

Values on generators

\((12,46)\) → \((-i,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 55 }(13, a) \) \(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(-i\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{3}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 55 }(13,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 55 }(13,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 55 }(13,·),\chi_{ 55 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 55 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2