Properties

Conductor 29
Order 28
Real No
Primitive No
Parity Odd
Orbit Label 58.f

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(58)
 
sage: chi = H[47]
 
pari: [g,chi] = znchar(Mod(47,58))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 29
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 28
Real = No
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = No
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = Odd
Orbit label = 58.f
Orbit index = 6

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 ]
 

\(\chi_{58}(3,\cdot)\) \(\chi_{58}(11,\cdot)\) \(\chi_{58}(15,\cdot)\) \(\chi_{58}(19,\cdot)\) \(\chi_{58}(21,\cdot)\) \(\chi_{58}(27,\cdot)\) \(\chi_{58}(31,\cdot)\) \(\chi_{58}(37,\cdot)\) \(\chi_{58}(39,\cdot)\) \(\chi_{58}(43,\cdot)\) \(\chi_{58}(47,\cdot)\) \(\chi_{58}(55,\cdot)\)

Inducing primitive character

\(\chi_{29}(18,\cdot)\)

Values on generators

\(31\) → \(e\left(\frac{11}{28}\right)\)

Values

-113579111315171921
\(-1\)\(1\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{17}{28}\right)\)\(i\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{19}{28}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{28})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 58 }(47,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{58}(47,\cdot)) = \sum_{r\in \Z/58\Z} \chi_{58}(47,r) e\left(\frac{r}{29}\right) = 3.5716618614+4.0302892635i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 58 }(47,·),\chi_{ 58 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{58}(47,\cdot),\chi_{58}(1,\cdot)) = \sum_{r\in \Z/58\Z} \chi_{58}(47,r) \chi_{58}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 58 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{58}(47,·)) = \sum_{r \in \Z/58\Z} \chi_{58}(47,r) e\left(\frac{1 r + 2 r^{-1}}{58}\right) = 7.7547355661+-2.7134993343i \)