Properties

Conductor 485
Order 32
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 485.bg

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(485)
 
sage: chi = H[67]
 
pari: [g,chi] = znchar(Mod(67,485))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 485
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 32
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 485.bg
Orbit index = 33

Galois orbit

sage: chi.sage_character().galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{485}(28,\cdot)\) \(\chi_{485}(52,\cdot)\) \(\chi_{485}(63,\cdot)\) \(\chi_{485}(67,\cdot)\) \(\chi_{485}(77,\cdot)\) \(\chi_{485}(117,\cdot)\) \(\chi_{485}(127,\cdot)\) \(\chi_{485}(142,\cdot)\) \(\chi_{485}(143,\cdot)\) \(\chi_{485}(148,\cdot)\) \(\chi_{485}(228,\cdot)\) \(\chi_{485}(263,\cdot)\) \(\chi_{485}(272,\cdot)\) \(\chi_{485}(333,\cdot)\) \(\chi_{485}(407,\cdot)\) \(\chi_{485}(443,\cdot)\)

Values on generators

\((292,296)\) → \((i,e\left(\frac{19}{32}\right))\)

Values

-112346789111213
\(1\)\(1\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{7}{8}\right)\)\(-i\)\(e\left(\frac{21}{32}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{19}{32}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{32})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 485 }(67,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{485}(67,\cdot)) = \sum_{r\in \Z/485\Z} \chi_{485}(67,r) e\left(\frac{2r}{485}\right) = 22.0094891363+0.7631434729i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 485 }(67,·),\chi_{ 485 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{485}(67,\cdot),\chi_{485}(1,\cdot)) = \sum_{r\in \Z/485\Z} \chi_{485}(67,r) \chi_{485}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 485 }(67,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{485}(67,·)) = \sum_{r \in \Z/485\Z} \chi_{485}(67,r) e\left(\frac{1 r + 2 r^{-1}}{485}\right) = 0.7101201441+3.5700150447i \)