Properties

Label 55.38
Modulus $55$
Conductor $55$
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(55, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([15,8]))
 
pari: [g,chi] = znchar(Mod(38,55))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 55.k

\(\chi_{55}(3,\cdot)\) \(\chi_{55}(27,\cdot)\) \(\chi_{55}(37,\cdot)\) \(\chi_{55}(38,\cdot)\) \(\chi_{55}(42,\cdot)\) \(\chi_{55}(47,\cdot)\) \(\chi_{55}(48,\cdot)\) \(\chi_{55}(53,\cdot)\)

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: 20.0.1402274470934209014892578125.1

Values on generators

\((12,46)\) → \((-i,e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\( \chi_{ 55 }(38, a) \) \(-1\)\(1\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(-i\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{7}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 55 }(38,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 55 }(38,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 55 }(38,·),\chi_{ 55 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 55 }(38,·)) \;\) at \(\; a,b = \) e.g. 1,2