Properties

Label 287.167
Modulus $287$
Conductor $287$
Order $8$
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(287, base_ring=CyclotomicField(8))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,3]))
 
pari: [g,chi] = znchar(Mod(167,287))
 

Basic properties

Modulus: \(287\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(8\)
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 287.l

\(\chi_{287}(27,\cdot)\) \(\chi_{287}(55,\cdot)\) \(\chi_{287}(167,\cdot)\) \(\chi_{287}(202,\cdot)\)

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: \(\Q(\zeta_{8})\)
Fixed field: 8.8.467605011588281.1

Values on generators

\((206,211)\) → \((-1,e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 287 }(167, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(-i\)\(e\left(\frac{7}{8}\right)\)\(i\)\(i\)\(-1\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 287 }(167,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 287 }(167,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 287 }(167,·),\chi_{ 287 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 287 }(167,·)) \;\) at \(\; a,b = \) e.g. 1,2