Properties

Conductor 388
Order 32
Real No
Primitive Yes
Parity Even
Orbit Label 388.s

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(388)
 
sage: chi = H[271]
 
pari: [g,chi] = znchar(Mod(271,388))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 388
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
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Even
Orbit label = 388.s
Orbit index = 19

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 ]
 

\(\chi_{388}(19,\cdot)\) \(\chi_{388}(51,\cdot)\) \(\chi_{388}(55,\cdot)\) \(\chi_{388}(63,\cdot)\) \(\chi_{388}(67,\cdot)\) \(\chi_{388}(127,\cdot)\) \(\chi_{388}(131,\cdot)\) \(\chi_{388}(139,\cdot)\) \(\chi_{388}(143,\cdot)\) \(\chi_{388}(175,\cdot)\) \(\chi_{388}(239,\cdot)\) \(\chi_{388}(263,\cdot)\) \(\chi_{388}(271,\cdot)\) \(\chi_{388}(311,\cdot)\) \(\chi_{388}(319,\cdot)\) \(\chi_{388}(343,\cdot)\)

Values on generators

\((195,5)\) → \((-1,e\left(\frac{7}{32}\right))\)

Values

-113579111315171921
\(1\)\(1\)\(e\left(\frac{13}{16}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{9}{32}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{5}{16}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{1}{32}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{7}{32}\right)\)\(e\left(\frac{3}{32}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 388 }(271,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{388}(271,\cdot)) = \sum_{r\in \Z/388\Z} \chi_{388}(271,r) e\left(\frac{r}{194}\right) = 0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 388 }(271,·),\chi_{ 388 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{388}(271,\cdot),\chi_{388}(1,\cdot)) = \sum_{r\in \Z/388\Z} \chi_{388}(271,r) \chi_{388}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 388 }(271,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{388}(271,·)) = \sum_{r \in \Z/388\Z} \chi_{388}(271,r) e\left(\frac{1 r + 2 r^{-1}}{388}\right) = 5.6782621889+28.5465517487i \)