Properties

Label 165.68
Modulus $165$
Conductor $165$
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(165, base_ring=CyclotomicField(20))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,15,2]))
 
pari: [g,chi] = znchar(Mod(68,165))
 

Basic properties

Modulus: \(165\)
Conductor: \(165\)
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 165.u

\(\chi_{165}(2,\cdot)\) \(\chi_{165}(8,\cdot)\) \(\chi_{165}(17,\cdot)\) \(\chi_{165}(62,\cdot)\) \(\chi_{165}(68,\cdot)\) \(\chi_{165}(83,\cdot)\) \(\chi_{165}(107,\cdot)\) \(\chi_{165}(128,\cdot)\)

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

Values on generators

\((56,67,46)\) → \((-1,-i,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\( \chi_{ 165 }(68, a) \) \(-1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 165 }(68,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 165 }(68,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 165 }(68,·),\chi_{ 165 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 165 }(68,·)) \;\) at \(\; a,b = \) e.g. 1,2