Properties

Label 960.731
Modulus $960$
Conductor $192$
Order $16$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(16))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,9,8,0]))
 
pari: [g,chi] = znchar(Mod(731,960))
 

Basic properties

Modulus: \(960\)
Conductor: \(192\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(16\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{192}(155,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 960.cn

\(\chi_{960}(11,\cdot)\) \(\chi_{960}(131,\cdot)\) \(\chi_{960}(251,\cdot)\) \(\chi_{960}(371,\cdot)\) \(\chi_{960}(491,\cdot)\) \(\chi_{960}(611,\cdot)\) \(\chi_{960}(731,\cdot)\) \(\chi_{960}(851,\cdot)\)

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: \(\Q(\zeta_{16})\)
Fixed field: 16.16.3965881151245791007623610368.1

Values on generators

\((511,901,641,577)\) → \((-1,e\left(\frac{9}{16}\right),-1,1)\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 960 }(731, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{7}{16}\right)\)\(i\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{11}{16}\right)\)\(1\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 960 }(731,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 960 }(731,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 960 }(731,·),\chi_{ 960 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 960 }(731,·)) \;\) at \(\; a,b = \) e.g. 1,2